# Time turners, Energy Conservation and General Relativity

This post is a bit of en­ter­tain­ment for sci­en­tifi­cally in­clined Harry Pot­ter fans.

Time turner from the Harry Pot­ter se­ries (and from the Eliezer Yud­kowsky’s ven­er­a­ble HPMoR fan­fic) is a very use­ful de­vice if you have some un­finished busi­ness in the re­cent past, like at­tend­ing an ex­tra class or sav­ing a friend from a cer­tain death. How­ever, Gen­eral Rel­a­tivity has a few words to say about them, and they are not very flat­ter­ing. I will only ad­dress one is­sue here: En­ergy con­ser­va­tion. TL;DR: if you use a time turner to van­ish into the past, those around you will see you blown to tiny bits of Mer­lin-knows-what, quickly dis­ap­pear­ing from view. When you ap­pear in the past, this ex­plo­sion ap­pears in re­verse.

Be­fore we get to the time turn­ers, how­ever, let us con­sider an aside.

Let us start with a com­mon ques­tion: if the Sun stop shin­ing this in­stant, when would we no­tice? The com­mon an­swer: it takes light 8.5 min­utes to travel the dis­tance of 150,000,000 km be­tween the Sun and the Earth, so that’s how long it will take. This glosses over the is­sue of what does “this in­stant” mean ex­actly at two differ­ent points in space, which is not so triv­ial given the rel­a­tivity of si­mul­tane­ity in Spe­cial Rel­a­tivity. It is eas­ily patched up, how­ever, once we fix a global frame of refer­ence. The Cos­mic Microwave Back­ground (CMB) is a nat­u­ral one to use, and both the Earth and the Sun travel with a neg­ligible frac­tion of the speed of light rel­a­tive to the CMB. Any­way, the an­swer is still very close to 8.5 min.

Now an­other, de­cep­tively similar ques­tion: if the Sun dis­ap­pears this in­stant, how long be­fore the Earth will stop or­bit­ing the point where it used to be? The com­mon an­swer: grav­ity trav­els with the speed of light, so also 8.5 min. This an­swer is ob­vi­ous, sim­ple and wrong. Yes, dead wrong. Why? be­cause static grav­ity is not like light, it’s more like elec­tric field, only worse.

Let’s first think of how you would make the Sun dis­ap­pear. Maybe it turned into a black hole? Well, this would not re­ally mean dis­ap­pear­ance of grav­ity, the mass of the black hole will still be that of the Sun, and the Earth will hap­pily (or un­hap­pily, as the case may be) con­tinue or­bit­ing the Sun’s corpse. So, in this case the an­swer is “it won’t stop or­bit­ing”.

OK, so black hole was a bad ex­am­ple. How about a worm­hole in­stead? You know, the evil Vo­gon-like aliens need to clear the room for a hy­per­space by­pass, and they build a worm­hole from far away and suck all the mat­ter in the Sun through it out of the way. What would hap­pen then? There are a cou­ple of hints: one is that from out­side a worm­hole is in­dis­t­in­guish­able from a black hole, and the other is the Gauss Law. Both hints lead one to the same an­swer: just like with turn­ing the Sun into a black hole, there is very lit­tle grav­i­ta­tional effect on the sur­round­ing space. The rest of the now ex-So­lar sys­tem will con­tinue mer­rily on its way around the point where our Sun used to be.

An aside for those cu­ri­ous about the Gauss Law ar­gu­ment. The law in its in­te­gral form states that the flux of the grav­i­ta­tional field in­ward through any closed sur­face en­com­pass­ing the Sun is pro­por­tional to the Sun’s mass. To change the field, you need to re­move some mass from in­side this imag­i­nary sur­face, by hav­ing it phys­i­cally cross the sur­face. This last point may not be ob­vi­ous, but it fol­lows from Gen­eral Rel­a­tivity. Speci­fi­cally, the Ein­stein’s most mi­s­un­der­stood the­ory says that the space­time cur­va­ture is de­ter­mined by the (past and pre­sent) dis­tri­bu­tion of mat­ter in space­time. There are some ex­cep­tions, like the fixed-mass spher­i­cal ob­jects, such as black holes and worm­holes, which con­tains no mat­ter, and grav­i­ta­tional ra­di­a­tion, which can carry away en­ergy. But if you take a spher­i­cal ob­ject like the Sun and try to calcu­late what hap­pens if you de­crease its mass, Gen­eral Rel­a­tivity tells you that this mass has gone out­ward from the Sun in all di­rec­tions in some form. It is not fussy about the form, as long as just the right amount of mass/​en­ergy has gone out.

Let me re­peat for those who skipped the above para­graph: if you take the Sun and de­crease its mass, the only way it can hap­pen if this mass leaves the Sun out­ward and dis­ap­pears into space. This hap­pens all the time, of course, the Sun con­stantly loses its mass through ra­di­a­tion and so­lar wind, or in more dras­tic cases through Su­per­nova ex­plo­sions. Effects like this prop­a­gate no faster than light, of course. So they take for­ever to prop­a­gate all the way to in­finity.

Now, back to the time turn­ers. Hermione Granger might be but a small if in­cred­ibly stu­dious girl, but she still has mass. If you were to peek at her us­ing a time turner and dis­ap­pear, her mass, small though it may be, still has to go some place, just like the dis­ap­pear­ing Sun’s mass had to go some place. The op­tions are few: she can blow into tiny pieces fly­ing past you, or dis­ap­pear in a flash of brilli­ant light (and it takes a lot of light to carry away 50kg, what’s with E=mc^2) . Ba­si­cally, it will not be a pretty sight. What can­not hap­pen is her sim­ply van­ish­ing, with no ill effects what­so­ever. Well, it can­not hap­pen if we are will­ing to keep Rel­a­tivity around. Maybe we don’t have to, what’s with a cer­tain deputy mis­tress turn­ing into a cat and back, prob­a­bly in­stantly chang­ing her mass, with no ill effects on her or her sur­round­ings. But if you give up on Gen­eral Rel­a­tivity, quite a few things will un­ravel, like all four New­ton’s laws.

Also don’t for­get the other side of the time turner ac­tion: Hermione ap­pear­ing out of thin air just be­fore walk­ing into her ex­tra class. The above pro­cess has to hap­pen in re­verse: an amount of mat­ter equiv­a­lent to her mass has to travel in­wards out of nowhere and co­a­lesce into a per­son. Where did this mat­ter come from? How did it form be­fore col­laps­ing into a per­son? How did it know that it would need to time its ar­rival into a cer­tain point perfectly with what­ever time turner will have been set to? That’s some hard-core magic right there. Also, suck it, the Se­cond law of Ther­mo­dy­nam­ics.

So, let me sum­ma­rize: mass can­not just dis­ap­pear, it has to spread out. mass can­not just ap­pear, it has to co­a­lesce. Thus time turn­ers can­not be used in­con­spicu­ously, ev­ery­one around would be well aware of one’s use, as­sum­ing they sur­vive it. Ac­tu­ally, it prob­a­bly can­not be used at all with­out break­ing Gen­eral Rel­a­tivity and/​or Ther­mo­dy­nam­ics. But hey, that’s what magic is for.

EDIT: this post cur­rently sits at −2 karma with 6 down­votes. I’d ap­pre­ci­ate if any of the peo­ple who thought “I want less of this” ex­pli­cate their logic to me, so I can do bet­ter next time.

EDIT2: OK, no one replied to my re­quest… I’m guess­ing that some of you guys just quietly hate me :)

• So what you’re say­ing is that in Soviet Mag­i­cal Bri­tain, WIMPs are for magic.

• We have already seen that HPMOR!magic can break the con­ser­va­tion of mass (Trans­figu­ra­tion of ob­jects of var­i­ous mass such as a can­teen of frozen wa­ter into a large rocket en­g­ine, An­i­ma­gus), con­ser­va­tion of mo­men­tum (’Ar­resto Mo­mento!, broom­sticks (ap­par­ently), telekinet­ics) etc.

• True, but those were lo­cal vi­o­la­tions. It was not stated that Gen­eral Rel­a­tivity must fail for magic to work, at least I don’t re­call it. Also an oc­ca­sional dis­cus­sion of Quir­rell com­mu­ni­cat­ing with one rather dis­tant ob­ject would look differ­ently if FTL is ex­plic­itly OK.

• What do you mean by lo­cal vari­a­tion. How do you know time turn­ers aren’t also just lo­cal vari­a­tions?

• In­ter­est­ingly, the boxes from the movie Primer can be made to avoid that prob­lem.

A short re­cap of how they work. You switch the box on, walk away from it to avoid run­ning into past you, come back to the box sev­eral hours later, switch it off, climb in­side, sit there for sev­eral hours, and climb out at the mo­ment the box was switched on. One rea­son this model is cool is that it avoids one com­mon prob­lem with fic­tional time travel, the chang­ing lo­ca­tion of the Earth. You don’t end up in in­ter­plane­tary space be­cause you travel back along the path of the box in space­time.

So here’s how you make the Primer boxes obey con­ser­va­tion of mass as well. The idea is that a box con­tain­ing a time-re­versed hu­man should weigh less than an empty box. Let’s say you weigh 70kg, and the box weighs 100kg when empty and switched off. When you switch the box on, a past ver­sion of you climbs out, and the box now weighs 30kg. Sev­eral hours later, a fu­ture you climbs in and the box now weighs 100kg, at which point the box switches off and sits there as empty as be­fore.

At first I felt pretty smart for figur­ing this out be­cause this whole is­sue never came up in the movie at all. And then I re­mem­bered the small de­tail that the boxes in the movie were an ac­ci­den­tal in­ven­tion, whose origi­nal pur­pose was to re­duce the mass of ob­jects. Wow.

That got me think­ing about the other pos­si­ble hole in the movie, namely all the aban­doned timelines. Can this model of time travel be made to work cor­rectly with not just space­time paths and con­ser­va­tion of mass, but also causal­ity and prob­a­bil­is­ti­cally branch­ing timelines? For ex­am­ple, if you travel back in time and kill your past self, can that yield a unique con­sis­tent as­sign­ment of prob­a­bil­ities to timelines, where all time trav­el­ers “come from some­where” and can’t af­fect their “prob­a­bil­ity weight”? The re­sult was this com­ment, for which I later found a proof of con­sis­tency which this mar­gin is too small to con­tain ;-)

• By the way, any­one who likes to think about magic-and-physics should prob­a­bly read Ra. Much like Harry, sci­en­tists find some­thing which looks like magic, rec­og­nize it’s a big deal, and pro­ceed to ap­ply sci­ence; but here the en­tire world is in on it, not a lone hero. Also, plenty of tech­ni­cal ex­po­si­tion (though less than if the au­thor weren’t try­ing to make it more char­ac­ter-fo­cused).

• Time turner from the Harry Pot­ter se­ries (and from the Eliezer Yud­kowsky’s ven­er­a­ble HPMoR fan­fic) is a very use­ful de­vice if you have some un­finished busi­ness in the re­cent past, like at­tend­ing an ex­tra class or sav­ing a friend from a cer­tain death. How­ever, Gen­eral Rel­a­tivity has a few words to say about them, and they are not very flat­ter­ing. I will only ad­dress one is­sue here: En­ergy con­ser­va­tion. TL;DR: if you use a time turner to van­ish into the past, those around you will see you blown to tiny bits of Mer­lin-knows-what, quickly dis­ap­pear­ing from view. When you ap­pear in the past, this ex­plo­sion ap­pears in re­verse.

Yes, the Time Turn­ers as de­scribed vi­o­late con­ser­va­tion of en­ergy. Some­thing is hap­pen­ing that doesn’t com­ply with Real­ity!Physics. Harry no­tices this the first time he en­coun­ters a witch. From Chap­ter 2:

“You turned into a cat! A SMALL cat! You vi­o­lated Con­ser­va­tion of En­ergy! That’s not just an ar­bi­trary rule, it’s im­plied by the form of the quan­tum Hamil­to­nian! Re­ject­ing it de­stroys uni­tar­ity and then you get FTL sig­nal­ling! And cats are COMPLICATED! A hu­man mind can’t just vi­su­al­ise a whole cat’s anatomy and, and all the cat bio­chem­istry, and what about the neu­rol­ogy? How can you go on think­ing us­ing a cat-sized brain?”

Pro­fes­sor McGon­a­gall’s lips were twitch­ing harder now. “Magic.”

“Magic isn’t enough to do that! You’d have to be a god!”

Pro­fes­sor McGon­a­gall blinked. “That’s the first time I’ve ever been called that.”

A blur was com­ing over Harry’s vi­sion, as his brain started to com­pre­hend what had just bro­ken. The whole idea of a unified uni­verse with math­e­mat­i­cally reg­u­lar laws, that was what had been flushed down the toi­let; the whole no­tion of physics. Three thou­sand years of re­solv­ing big com­pli­cated things into smaller pieces, dis­cov­er­ing that the mu­sic of the planets was the same tune as a fal­ling ap­ple, find­ing that the true laws were perfectly uni­ver­sal and had no ex­cep­tions any­where and took the form of sim­ple maths gov­ern­ing the small­est parts, not to men­tion that the mind was the brain and the brain was made of neu­rons, a brain was what a per­son was -

And then a woman turned into a cat, so much for all that.

In both the case of Trans­mo­grifi­ca­tion and Time Turn­ing ei­ther Con­ser­va­tion of En­ergy is part of a the­ory which just is not de­scribing re­al­ity or there is some mass re­dis­tri­bu­tion that Harry just hasn’t de­tected. For ex­am­ple, con­ver­sion of ex­cess mass into some kind of Dark Mat­ter that ex­ists in the MoR!Uni­verse but nei­ther wiz­ards or physi­cists have stud­ied yet.

Given the pos­si­bil­ity of a Trans­mo­grifi­ca­tion mechanism and ei­ther any ca­pa­bil­ity for send­ing in­for­ma­tion back in time or the pres­ence of an ex­ces­sively pow­er­ful pre­dic­tor prior to the ‘ear­liest’ end of the time jump, Time Turn­ing (or the ap­pear­ance thereof to all ob­servers) doesn’t in­tro­duce any ad­di­tional in­san­ity.

Also don’t for­get the other side of the time turner ac­tion: Hermione ap­pear­ing out of thin air just be­fore walk­ing into her ex­tra class. The above pro­cess has to hap­pen in re­verse: an amount of mat­ter equiv­a­lent to her mass has to travel in­wards out of nowhere and co­a­lesce into a per­son. Where did this mat­ter come from?

Pre­sum­ably the same place that the mat­ter came from when MoR!McGon­na­gal!cat turned back into MoR!McGon­na­gal!hu­man—ei­ther as yet un­de­tected lo­cal con­ver­sion or from Magic.

How did it form be­fore col­laps­ing into a per­son?

Tricky ques­tion, which gen­er­al­ises to the gen­eral prob­lem of de­ter­min­ing what all of this ‘magic’ stuff re­duces to or is based on. Si­mu­la­tion, ad­vanced de­cep­tion or ex­tremely pow­er­ful tech­nol­ogy from an over­whelm­ingly su­pe­rior civ­i­liza­tion or agent seem to be the ob­vi­ous hy­pothe­ses. (That is, con­di­tion­ing on the ob­ser­va­tions be­ing ac­cu­rate and not the re­sult of in­san­ity. Elimi­nat­ing that as the most plau­si­ble hy­poth­e­sis would take a heck of a lot of ev­i­dence.)

How did it know that it would need to time its ar­rival into a cer­tain point perfectly with what­ever time turner will have been set to?

Similar prob­lem and (at least some) ad­di­tional ev­i­dence for there be­ing a pow­er­ful agent at work in one of the var­i­ous ways that could oc­cur.

That’s some hard-core magic right there. Also, suck it, the Se­cond law of Ther­mo­dy­nam­ics.

In­deed. In fact one could even go so far as to in­clude that as part of the de­scrip­tion of the gen­res of Science Fic­tion and Fan­tasy.

• Those who think they un­der­stand Gen­eral Rel­a­tivity might warm up to the task of ret­con­ning Time-Turn­ers by solv­ing the fol­low­ing co­nun­drum, which takes place in the real world, and so must be solved un­der the con­straint that “magic” is not an al­lowed an­swer.

The Earth or­bits at 18 miles per sec­ond, and causal in­fluences from the Sun travel no faster than light, which takes 8 min­utes to reach the Earth from the Sun. There­fore the Earth “sees” the Sun 18x8x60 miles be­hind the straight line join­ing the two (us­ing a Sun-cen­tred frame of refer­ence). This is an an­gle to the ra­dial line of 18x8x60/​93000000 = 0.0001 ra­di­ans, or about 13 of a minute of an­gle. The grav­i­ta­tional at­trac­tion, be­ing a causal in­fluence trans­mit­ted at light­speed, must be along that line also, thus re­tard­ing the Earth in its or­bit with an ac­cel­er­a­tion of 0.0001 times the ra­dial ac­cel­er­a­tion. This is enough to can­cel out the Earth’s or­bital ve­loc­ity in about 1000 years.

This has not hap­pened. Where is the er­ror?

• Teach the he­lio­cen­tric con­tro­versy!

• The Earth re­volves around the point where the sun was 8 min­utes ago. In a so­lar-cen­tric frame of refer­ence, this is where the sun is now. In any other frame of refer­ence, “8 min­utes” has changed.

• I don’t know the rel­a­tivis­tic math, but: The Sun is not act­ing on the Earth; it would act iden­ti­cally on any mass that hap­pens to be any­where along that or­bit. To use your words, “The grav­i­ta­tional at­trac­tion, be­ing a causal in­fluence trans­mit­ted at light­speed, must be along that line”, but it is on ev­ery other line in­ter­sect­ing the Sun as well. There­fore light­speed is not a con­straint on what hap­pens.

• I’m pretty sure there’s no gen­eral rel­a­tivity in the er­ror:

Draw the tri­an­gle from the earth to the sun to the point where the earth will be when it in­ter­cepts light from the sun that is be­ing emit­ted ‘now’. The fu­ture Earth that is see­ing the light from the sun now is see­ing the light that the Earth emit­ted along the hy­potenuse, which is not at the same di­rec­tion a the Earth is ‘now’, but rather is in the di­rec­tion from the sun to where the Earth will be when it in­ter­cepts the light. In other words, I dis­agree that the Earth ‘sees’ the sun in the wrong spot.

• Why are you us­ing the Sun as the frame of refer­ence when ask­ing what the Earth is see­ing? (I’m not sure if this is the er­ror you’re look­ing for or not, but it strikes me as in­tu­itively wrong.)

• Be­cause it’s an in­er­tial frame, to the ac­cu­racy re­quired, but the Earth’s frame isn’t.

Mind you, I can’t claim to know any­thing about GR, cer­tainly not by this stan­dard. My ex­pec­ta­tion is that when one does the math­e­mat­ics, GR does not pre­dict planets spiral­ling to de­struc­tion on thou­sand-year timescales, and that shminux has lo­cated the prob­le­matic step. But I do not know the math­e­mat­ics. My ex­pec­ta­tion is also that ver­bal sto­ries to jus­tify why it does or does not hap­pen are only helpful once one has worked out the real math­e­mat­i­cal story, and can­not by them­selves re­solve the mat­ter.

• Ac­tu­ally the is­sue is the state­ment “The grav­i­ta­tional at­trac­tion, be­ing a causal in­fluence trans­mit­ted at light­speed”.

• Umm, what? I stud­ied GR way less than you, but so far as I know, there’s noth­ing even slightly prob­le­matic about that par­tic­u­lar phrase. Per­haps you’re refer­ring to its failure to im­ply the phrase af­ter it? Be­cause the fol­low­ing phrase is definitely wrong.

• “The grav­i­ta­tional at­trac­tion, be­ing a causal in­fluence trans­mit­ted at light­speed” is wrong here be­cause static fields do not prop­a­gate, they are just there. That’s why black hole can re­tain elec­tric charge and mass, even though “not even light can es­cape it”. EM and grav­i­ta­tional waves prop­a­gate at the speed of light, but that’s not re­lated to grav­i­ta­tional at­trac­tion or the coulomb force be­tween bod­ies. But you know all that. So it’s a wrong premise which mo­ti­vates a wrong con­clu­sion, which is the next phrase.

• I won’t say any­thing about grav­i­ta­tional waves, but in EM, what you just said is un­re­lated, to­tally is re­lated. If you just take the rules for elec­tro­stat­ics and the mag­netic field from charges, and ap­ply time-re­tar­da­tion, you re­cover the elec­tro­mag­netic waves pro­duced by that charge’s mo­tion. No cor­rec­tions or ad­just­ments are needed.

The elec­tri­cal and mag­netic fields at a given point is pre­cisely the sum over all charged par­ti­cles of their mo­men­tary elec­tri­cal and mag­netic fields at that point, causally offset… (… plus any back­ground fields that the uni­verse was shipped with, that did not arise from charges. I’m not say­ing that that’s im­pos­si­ble)

But the point is, it’s to­tally com­pletely fair to talk about causal in­fluences of a charged par­ti­cle or mass prop­a­gat­ing at light­speed. If a par­ti­cle’s sit­ting there and then gets kicked and set­tles down, the elec­tro­static and grav­i­ta­tional fields up­date to their new val­ues by the par­ti­cle emit­ting waves.

• I won’t say any­thing about grav­i­ta­tional waves, but in EM, what you just said is un­re­lated, to­tally is re­lated. If you just take the rules for elec­tro­stat­ics and the mag­netic field from charges, and ap­ply time-re­tar­da­tion, you re­cover the elec­tro­mag­netic waves pro­duced by that charge’s mo­tion. No cor­rec­tions or ad­just­ments are needed.

That’s true, but that’s waves, not static at­trac­tion. In essence, elec­tric at­trac­tion is “mag­i­cally” cor­rected for the straight-line mo­tion, so the elec­tric field from a mov­ing but non-ac­cel­er­at­ing charge points ex­actly in the di­rec­tion of the charge, not in the di­rec­tion where the charge was af­ter ac­count­ing for time-re­tar­da­tion. This breaks down once you add ac­cel­er­a­tion, hence EM waves. The origi­nal post made this (pos­si­bly de­liber­ate) mis­take: calcu­lated re­tarded field for (nearly) uniform mo­tion when calcu­lat­ing the di­rec­tion of the at­trac­tive force. Un­less I mi­s­un­der­stood it.

For grav­ity the cor­rec­tions re­sult­ing in ra­di­a­tion ap­pear even later, its “pre­dic­tive power” is one or­der in time deriva­tives bet­ter than that of EM.

• That’s true, but that’s waves, not static at­trac­tion. In essence, elec­tric at­trac­tion is “mag­i­cally” cor­rected for the straight-line motion

Wat. This is so severely coun­ter­in­tu­itive I’m go­ing to have to look it up and get a tech­ni­cal ex­pla­na­tion. Is there a named effect for this?

• I don’t re­call the name, but here is a neat java ap­plet vi­su­al­iz­ing the situ­a­tion. In essence, for uniform mo­tion the field lines are always straight, point­ing away from the charge, while the di­rec­tion of light re­ceived from the charge points to the re­tarded po­si­tion. There is a stan­dard de­tailed calcu­la­tion (like the one in Griffiths or Jack­son) here, with the fol­low­ing con­clu­sion:

which con­firms that the elec­tric field at R points along the di­rec­tion from R to the pre­sent (not the re­tarded) po­si­tion of the charge.

• Very in­ter­est­ing...

But it still doesn’t change the causal re­la­tion­ship: that al­ign­ment only ap­plies if noth­ing hap­pens to the other thing in the time since the cen­ter of that re­tarded cone. If no new in­for­ma­tion has been gen­er­ated, then sure, you can use the new in­for­ma­tion in­stead of the old in­for­ma­tion. But if any­thing hap­pens, you had bet­ter use the old in­for­ma­tion!

To get more for­mal about it: Con­sider or charge (equiv­a­lently, a mass) whose wor­ldline co­in­cides with (t, 0,0,0) for all t ⇐ 0) in some refer­ence frame

The field at the event (10, 10, 0, 0 ) oc­curs af­ter (0,0,0,0) in ev­ery sub­lu­mi­nal refer­ence frame.

The field at (10, 10, 0, 0 ) is in­de­pen­dent of what­ever hap­pens at (1, 0,0,0). If a laser comes in and knocks that charge aside, there’s zero differ­ence. None at all.

Sup­pose the charge was deflected so that it passes through (10, 1, 0, 0). You can’t get the elec­tri­cal field at event (10, 10, 0, 0) by look­ing at what the charge is do­ing at (10, 1, 0, 0) - you need to look at (0,0, 0,0).

This is what causal­ity looks like. The causal in­fluences prop­a­gate at light­speed. Even elec­tro­static ones.

• But it still doesn’t change the causal re­la­tion­ship: that al­ign­ment only ap­plies if noth­ing hap­pens to the other thing in the time since the cen­ter of that re­tarded cone.

Ab­solutely. As I said

This breaks down once you add ac­cel­er­a­tion, hence EM waves.

How­ever I am not sure I agree with the part in bold:

This is what causal­ity looks like. The causal in­fluences prop­a­gate at light­speed. Even elec­tro­static ones.

When you say “Sup­pose the charge was deflected”, you have bro­ken the elec­tro­static as­sump­tions, since the charge is now ac­cel­er­at­ing. Depend­ing on the dis­tance, you ei­ther get the near-field effects or the ra­di­a­tive effects, which do in­deed prop­a­gate at light­speed. Once the ac­cel­er­a­tion dis­ap­pears and the light-speed tran­sients died down, you are back in the elec­tro­static/​mag­ne­to­static mode with lag-free fields.

• That was the point of the ex­am­ple—by time 10, the charge was no longer ac­cel­er­at­ing, but you know that you’re not clear to use elec­tro­stat­ics be­ing ‘non­causal’ yet be­cause the light cone hasn’t reached that far. Be­ing lag free is a com­pu­ta­tional con­ve­nience that some­times ap­plies, and you need to know when by ap­ply­ing causal­ity.

So, it is ALWAYS fair to say that fields are causal in­fluences, whether they’re static or dy­namic. That was why I ob­jected to your com­plaint in the first place.

• Isn’t “the di­rec­tion of the charge” where you point a telescope to look at it, even if it is mov­ing?

• Not nec­es­sar­ily. This is some­what counter-in­tu­itive, but the light lags the di­rec­tion of the Coulomb’s law’s at­trac­tion if a charge moves past you with a con­stant ve­loc­ity (and has been do­ing so for some time). The at­trac­tion force points to the “true” di­rec­tion of the charge, whereas light takes a bit to catch up.

Note that this ap­par­ent FTL effect can­not be used to trans­mit any in­for­ma­tion FTL, be­cause, as soon as you try to wig­gle the charge to tele­graph some­thing, this wig­gling will only be sent as EM ra­di­a­tion (light), at light speed.

• So, it’s pos­si­ble for the elec­tri­cal force to act from a lo­ca­tion that never had a charge? If the charge moves at a con­stant speed long enough and then makes a hard turn away, the charge will act on other ob­jects (at least briefly) as if the charge had con­tinued straight? Does the charge also act on the ob­ject as though it had not turned, even af­ter it had, or is the force unilat­eral?

• Yes to the sec­ond, not sure how the third is differ­ent. That’s why, in part, New­ton’s sec­ond law does not in gen­eral hold for Elec­tro­mag­netism, but mo­men­tum con­ser­va­tion does, if you ac­count for the mo­men­tum of the elec­tro­mag­netic field it­self.

• That be­hav­ior is con­trary to naive ex­pec­ta­tions, right? If I run re­ally fast to­wards a wall but turn be­fore I reach it, I shouldn’t hit it. It also shouldn’t smash my face in af­ter I make the turn.

The ma­jor force in­volved in billiard balls bounc­ing off of each other is elec­tric in na­ture, right?

• Right, ev­ery­one knows a free body just moves in a straight line.

• This is noth­ing com­pared to the im­pos­si­bil­ity of Time-Turn­ers given MWI, which is of course a given. I’ve been as­sum­ing that HPMOR runs on col­lapse QM.

• What if a time turner func­tions like this:

1. The Time Turner, un­known to you and hours prior to any spin­ning, cre­ates a near-du­pli­cate of you (and it­self) some­where in the world

2. The Time Turner con­trives for you to end up end up ex­actly like the du­pli­cate it cre­ated, n hours later, and then com­pels you to turn it­self (like ComedTea com­pels you to drink it).

3. On be­ing turned, the TIme Turner de­stroys you and it­self.

• I’ve been as­sum­ing that HPMOR runs on col­lapse QM.

Another at­tempt to re­duce the prob­a­bil­ity mass of this fan­tasy world?

• I think it fits best with a trans­ac­tional form, ac­tu­ally. Time-turn­ers just provide al­ter­nate routes for the trans­ac­tions over their length, and you get a self-con­sis­tent out­come.

Hmm. Now I want to see what hap­pens if you try to do a quan­tum in­terfer­ence ex­per­i­ment in­volv­ing a time-turner. Good luck get­ting the phase to stay co­her­ent over an hour, in­clud­ing the pro­cess of time-turn­ing, though.

• Now I want to see what hap­pens if you try to do a quan­tum in­terfer­ence ex­per­i­ment in­volv­ing a time-turner.

My pre­dic­tion: “DO NOT MESS WITH TIME”

• Hmm, I don’t see how MWI would be in­terfer­ing with time-turn­ers. If any­thing, you can model a time-turner as spawn­ing a new world at the des­ti­na­tion point in the past, if you are will­ing to over­look the mass non-con­ser­va­tion is­sue. Un­wind­ing a di­rected cyclic graph into a tree would also re­solve the grand­father para­dox and the “don’t mess with time” mes­sage-in-a-bot­tle.

• Isn’t “re­solv­ing the grand­father para­dox” the whole rea­son for the im­pos­si­bil­ity of Time-Turn­ers (as op­posed to generic time ma­chines)? HP and HPMoR time travel seems to be of the Novikov va­ri­ety, which isn’t what you’d ob­serve if trips into the past aren’t con­strained to re­pro­duce pre­vi­ously-ob­served his­tory. In HP you can’t kill your grand­father be­cause you didn’t kill your grand­father (ex­act mechanism to be hand­waved later). In MWI/​spawn­ing time travel you can kill your grand­father, and then that new his­tory just won’t pro­duce an­other baby you.

• Right. In gen­eral, spawn­ing time travel is para­dox-free, that’s why I am not clear on why “the im­pos­si­bil­ity of Time-Turn­ers given MWI”. Pre­sum­ably if you already spawn un­countable num­bers of wor­lds all the time, it’s not a big deal to spawn one more.

• You can cer­tainly pos­tu­late a physics that’s both MWI and con­tains some­thing sorta like Time-Turn­ers ex­cept with­out the Novikov prop­erty. The prob­lem with that isn’t para­dox, it just doesn’t re­pro­duce the fic­tional ex­per­i­men­tal ev­i­dence we’re try­ing to ex­plain. What’s im­pos­si­ble is MWI with some­thing ex­actly like Time-Turn­ers in­clud­ing Novikov.

• (Nods.)

• What’s im­pos­si­ble is MWI with some­thing ex­actly like Time-Turn­ers in­clud­ing Novikov.

I am ig­no­rant on these top­ics, but isn’t Novikov con­sis­tency pred­i­cated on QM? In that the “ac­tual” para­dox-free world is pro­duced by a sum-over-his­to­ries? What about MWI pre­vents this?

Sorry if this is an in­cred­ibly stupid ques­tion.

• Novikov con­sis­tency is syn­ony­mous with Stable Time Loop, where all time trav­el­ers ob­serve the same events as they re­mem­ber from their sub­jec­tively-pre­vi­ous iter­a­tion. This is as op­posed to MWI-based time travel, where the no para­dox rule merely re­quires that the over­all dis­tri­bu­tion of time trav­el­ers ar­riv­ing at t0 is equal to the over­all dis­tri­bu­tion of peo­ple de­part­ing in time ma­chines at t1.

Yes, Novikov talked about QM. He used the sum-over-his­to­ries for­mu­la­tion, re­stricted to the sub­set of his­to­ries that each sin­gle­hand­edly form a clas­si­cal sta­ble time loop. This al­lows some form of mul­ti­ple wor­lds, but not stan­dard MWI: This for­bids any Everett branch­ing from hap­pen­ing dur­ing the time loop (if any event that af­fects the time trav­eler’s state branched two ways, one of them would be in­con­sis­tent with your mem­ory), and in­stead branches only on the ques­tion of what comes out of the time ma­chine.

• Hmm. So if, say, I com­mit­ted quan­tum suicide, then trav­eled back, I wouldn’t have any spe­cial in­for­ma­tion about the re­sult of the RNG. Most of me would still end up in wor­lds where I died; God’s dice get re-rol­led ev­ery time round. No ex­tra math to pre­vent para­doxes; al­though it still looks like Novikov for non-quan­tum events.

Whereas un­der stan­dard Novikov Con­sis­tency, I’m re­stricted to the wor­lds where I sur­vived, be­cause oth­er­wise I came from nowhere. In fact, the uni­verse is re­stricted to those wor­lds; there are only wor­lds where I sur­vived and came back and wor­lds where I died and didn’t. Thus, no Everett branch­ing. Right.

The de­gree to which the differ­ence would be ob­serv­able de­pends on the amount of quan­tum var­i­ance in your life, I guess.

• From sci­ence fic­tion physics[1]: Could the con­ser­va­tion laws be ex­panded so all times are in­cluded?

[1] I don’t re­mem­ber where I saw the no­tion, but pos­si­bly Hein­lein or Poul An­der­son.

• Could the con­ser­va­tion laws be ex­panded so all times are in­cluded?

This is in­deed the stan­dard ap­proach in mod­ern physics, yes. But then it gets even weirder.

Sup­pose that the per­son ac­ti­vat­ing the time turner is turned into a blob of neu­trinos which ra­di­ate in all di­rec­tions un­de­tected. Or just stayed where the per­son used to be, some­how. But they still have to as­sem­ble into this per­son some time in the past, so they can­not be or­di­nary neu­trinos, they have to be the spe­cial trav­el­ing-back-in-time ones. Now, in the usual par­ti­cle physics a par­ti­cle trav­el­ing back in time is equiv­a­lent to an an­tipar­ti­cle trav­el­ing for­ward in time. Same mass, but (nearly) ev­ery other nu­mer­i­cal quan­tum prop­erty re­versed. In this case even their mass has to be nega­tive.

So, here is a workaround. When Hermione (or Harry in HPMoR) ap­pears in the past, an un­de­tectable ghost of equal and op­po­site mass sep­a­rates from her, leav­ing the to­tal mass un­changed (zero). While she goes about her busi­ness, this in­visi­ble nega­tive mass ghost-Hermione gets to the fi­nal des­ti­na­tion in the fu­ture when and where she ac­ti­vated her time turner in or­der to go back, then an­nihilates the poor girl. Well, the “real” Hermione is now the other one, so that’s OK. Voila, en­ergy is con­served. If you don’t mind nega­tive-mass ghosts walk­ing around for a time.

• Has it been es­tab­lished whether time turn­ers can re­turn some­one to be­fore the time turner was cre­ated? If they can­not, then it is sim­ple to pos­tu­late that the time turner brings into ex­is­tence a perfect copy of the user at the time and lo­ca­tion of ar­rival, and then de­stroys the user at the time of de­par­ture. The time turner it­self would have to con­tain enough en­ergy to cre­ate the user to con­serve en­ergy.

That would also mean that a mass/​en­ergy con­serv­ing time turner has less stored magic dur­ing the pe­ri­ods of time when a du­pli­cate ex­ists.

Timeline, from the point of view of the two time turn­ers, refer­enc­ing the time on the clocks:
Time turner is cre­ated and be­comes fully charged. (Long time ago)
Copy of user and new de­pleted time turner cre­ated, de­plet­ing charge on time turner. (1200 noon)
User and time turner de­stroyed by use, recharg­ing new time turner. (1300)

From the point of view of the new time turner:
Mag­i­cally cre­ated in a de­pleted state along with copy of user. (1200 noon)
Recharged by the de­struc­tion of the origi­nal time turner and user (1300)

This might not mean that one can­not ac­quire a time turner, trade it with one’s fu­ture self’s time turner, then use that time turner to travel back and trade with one’s past self. The timelines would look like:
Man­u­fac­tured
Created du­pli­cate user and du­pli­cate time turner.
(Ex­changed)
Recharged by de­struc­tion of user and du­pli­cate time turner.
and:
Created along with du­pli­cate user
(Ex­changed)
De­stroyed along with origi­nal user to recharge origi­nal turner

A slightly more com­plex vari­a­tion al­lows the time-tuner to con­tain mul­ti­ple charges and be­have in the same man­ner as !Harry uses it. An ad­di­tional fac­tor is re­quired to ex­plain the 8-hour limi­ta­tion of mul­ti­ple time turn­ers.

• It’s 6 hours, i think, re­gard­less of the chain of time-turn­ers)

And given the 6-hour limi­ta­tion, all you have to posit is a 6-hour rit­ual for cre­at­ing the time turner be­fore it’s use­able. presto, no go­ing back to be­fore the time-turner was cre­ated.

...but the BIG time ma­chine in the ministry may or may not com­ply with the con­straints of the smaller time turn­ers. lastly, the 6-hour limi­ta­tion it­self… seems more like a way to premp­tively pre­vent a harry vs. quir­re­mort TIME WAR than any­thing else.

• Has it been es­tab­lished that, in ad­di­tion to never be­ing able to re­treat more than 6 hours us­ing any com­bi­na­tion of !time turn­ers, you can­not use more than one !time turner to travel back to the same hour re­peat­edly? I see many munchkin ways to abuse ei­ther pos­si­bil­ity (one be­ing to use mul­ti­ple time-turn­ers to be in an ar­bi­trary num­ber of places at the same time, and the other to take a tick back mul­ti­ple times and then plant it on the op­po­nent.

• yes, it has. when harry was try­ing to stun Moody, moody asked harry if “are you go­ing to give up, or do you think you’re go­ing to win?” Harry replied that he was on his last copy be­cause he had used up his re­main­ing hours study­ing.
al­though...hang on, is it 6 or 7?

Any­ways. Eliezer nerfed the time-turner pre­ci­cely to pre­vent Harry from ut­terly to­tally munck­in­ning it. “pre­vent or limit time­travel-based gamibt SPAM” and all that.

• Is that an in­her­ent limit, or sim­ply the re­sult of !Harry hav­ing ac­cess to only one time turner?

• come to think of it, it might not be a full-blown limi­ta­tion… But “time does not like be­ing stretched”. Even if it IS pos­si­ble to squeeze one ex­tra hour in by us­ing a sec­ond time turner, it’s not a good idea. (side effects may in­clude in­san­ity, ex­tremely-rapid ag­ing (more so than liv­ing 8 ex­tra hours in a day would lead you to ex­pect), and dizzi­ness.)

• If time is an­thopic enough to like things in a literal sense, it can be bar­gained with. That might be the prin­ci­ple of op­er­a­tion of the time-turn­ers.

If time can be bar­gained with, a ra­tio­nal­ist who em­braces a de­ci­sion the­ory that one-boxes when the boxes are trans­par­ent has a lot of lev­er­age.

• Ok, I went and found Where the quote came from. chap­ter 17...

“Be­cause it’s pretty im­pres­sive if you’re do­ing all that on just thirty hours a day.” There was an­other slight pause, [snip] “I’m afraid Time doesn’t like be­ing stretched out too much”

I think this is more a case of Dum­ble­dore us­ing an­thropic short­hand than ac­tual an­thropic rea­son­ing. He seems to have...(ooh, here’s a ran­domish the­ory: Dum­ble­dore ac­ci­den­tally(?) performed a rit­ual that sac­ri­ficed a few pieces of his over­all san­ity in or­der to pro­tect him­self against dan­ger­ous time-travel re­lated phe­nom­ena..)

Now where was I?...

Oh yeah. So that very much looks like ex­tra time-clones MIGHT be pos­si­ble with stan­dard time-turn­ers, but it is NOT a good idea.

Now, the Big Time Turner in the ministry (The one which had the ever hatch­ing/​un-hatch­ing egg) prob­a­bly COULD give you time-clone armies, laser-guided peggy-sues, and pos­si­bly even “change time” (what­ever that means), but it is prob­a­bly even more dan­ger­ous to use-you’d ba­si­cally have to cre­ate your own refer­ence frame, or else un­der­stand the real rules of Time.

But...maybe, just maybe: The BTT+philoso­phers stone+re­s­ur­rec­tion stone...

• To cre­ate a user from noth­ing, the time turner would have to ac­quire nega­tive mass equal to that of the trans­ported user. To de­stroy the user at the mo­ment of trans­port, the time turner would have to have had nega­tive mass prior to the use. In short, user mass + time-turner mass = 0 at all times. Which would mean that a time turner is much more use­ful as a lev­i­ta­tion aid.

In gen­eral, nega­tive mass be­haves weirdly. The New­ton’s laws still ap­ply, so the grav­i­ta­tional force act­ing on the nega­tive-mass time turner is mg (up­wards if m < 0). How­ever, if you let it go, it will fall down, not up (F=ma, so a=g, since m can­cels). If you push on a nega­tive-mass ob­ject, it tends to ac­cel­er­ate to­ward you. This makes car­ry­ing one quite haz­ardous: you can only pull it (in which case it gets pushed). If you even touch it, it will snap your hand.

Given the above, I think it is safe to say that time turn­ers have pos­i­tive (and small) mass and don’t change their mass when ac­ti­vated.

• To cre­ate a user from noth­ing, the time turner would have to ac­quire nega­tive mass equal to that of the trans­ported user. To de­stroy the user at the mo­ment of trans­port, the time turner would have to have had nega­tive mass prior to the use.

No it doesn’t.

The time-turner just has to con­tain more mass-en­ergy than the user at all times. Pretty sim­ple, re­ally. See it as a mag­i­cal con­tainer and beamer for all that mass that gets dis­placed (and while cre­at­ing the user-copy or delet­ing it, it also re­moves/​gen­er­ates an ap­pro­pri­ate amount of air to equal­ize the pres­sure effects).

The pro­cess would, by the clock, look like this:

• Time-turner is cre­ated, has a mass of 5000 (the mass is not ap­par­ent for the same rea­sons that wingardium leviosa makes ob­jects float /​ less heavy, whichever rea­sons those might be).

• Time-turner re­moves some air (mass 5000 + 2) and si­mul­ta­neously gen­er­ates a copy of the user (mass 5002 − 80)

• Time-turner later re­moves the “old” /​ time-turned user (mass 4922 + 80) and gen­er­ates some air to fill in the space (mass back to 5000).

As for how the time-turner it­self moves places, well, it pre­sum­ably ei­ther shares the mass among copies some­how (en­tan­gle­ment? worm­holes? ex­otic space ge­ome­tries?) or just tele­ports the mass re­serve where ap­pro­pri­ate, or per­haps the “en­chant­ment” is con­tained in a sep­a­rate “server” of sorts (which would be most co­her­ent with the At­lantis hy­poth­e­sis).

This ex­pla­na­tion con­serves mass—the mass is just taken from a big rock or some­thing at the cre­ation of the time-turner, and then re­dis­tributed/​re­filled as ap­pro­pri­ate to cre­ate the ap­par­ent effect of cre­ation-from-noth­ing.

• I agree that if you al­lowed to screen a fixed amount of en­ergy from GR by magic, you don’t have to have nega­tive en­ergy, as you can add an ar­bi­trary and un­de­tectable amount of it to what­ever ob­ject you want. But then why stop there, GR is already bro­ken.

• GR was (ap­par­ently) bro­ken by changes in mass; sup­pose in­stead that an­i­magi con­vert be­tween magic, en­ergy, and mass in an analo­gous man­ner: When in a smaller form, they are more mag­i­cal, and vice versa. Brooms con­vert magic to a com­bi­na­tion of en­ergy and/​or some­thing which is anti-mass but not anti-in­er­tia, and/​or some­thing else which is not di­rectly analo­gous to any­thing in GR.

The time turner then re­mains no harder than ap­pa­ra­tion com­bined with fu­ture pre­dic­tion; fu­ture pre­dic­tion is easy enough that it’s pro­vided in nov­elty drinks.

• or per­haps the “en­chant­ment” is con­tained in a sep­a­rate “server” of sorts (which would be most co­her­ent with the At­lantis hy­poth­e­sis)

This also pro­vides a pos­si­ble mechanism for the 8-hour limit on mul­ti­ple time turn­ers; the servers can pre­dict up to eight hours in the fu­ture, ex­cept they can­not ac­cu­rately pre­dict their own fu­ture pre­dic­tion. Things which do not de­pend on their fu­ture pre­dic­tion can mostly still be pre­dicted “Do not mess with time” could have origi­nated from there as well.

• To say that some­thing is con­served means that it is the same at one time as it is at an­other time.

If you can­not time travel, and you have a set of ob­jects at 1 PM, then you can com­pare them to the same set of ob­jects at an­other time, such as an hour later.

If you can time travel, you can do the same—but terms such as “at an­other time” and “an hour later” be­come tricky.

With time travel, the time as mea­sured in the set­ting is not the same as the time as mea­sured by the ob­jects them­selves. If some of those ob­jects time travel from 2:30 back to 1:30, then at 2 PM, 1 hour passed in the set­ting, you have some ob­jects for which 1 hour has passed, and you have other ob­jects for which 2 hours have passed. If you add those ob­jects up just be­cause 1 hour has passed in the set­ting you are adding the wrong things. You should be adding the ob­jects for which one hour has passed (ig­nor­ing the ob­jects for which two hours have passed) be­cause con­ser­va­tion means that the amount is pre­served one hour later—you can’t mix ob­jects from one hour later and from two hours later and ex­pect to get an amount that is con­served. The fact that one hour has passed in the set­ting is use­ful in the non time travel case be­cause the time that has passed in the set­ting is the same as the time that has passed for all of the ob­jects. In the time travel case the fact that one hour has passed in the set­ting is just a dis­trac­tion.

• None of what you wrote makes any sense to me, sorry.

• Here’s how I read it:

Mass is nei­ther cre­ated nor de­stroyed. It is moved.

As­sum­ing you can tell if en­ergy is con­served by check­ing if it’s still there later, when you have a time ma­chine, is like claiming a bolder has vi­o­lated mass/​en­ergy con­ser­va­tion be­cause some­body broke a piece off.

• The whole point is that in GR con­ser­va­tion is lo­cal, you should be able to trace the move­ment of mass through space.

• in GR con­ser­va­tion is local

Ex­actly. You were treat­ing it as gen­eral.

• Aside from the ob­vi­ous ‘It’s magic, what are you talk­ing about’ com­plaint, I’d like to point out that if you have a worm­hole and you draw a Gaus­sian sur­face around it, you’re not done—you also need to draw a sur­face on the other side of the worm­hole. Note that these will have equal and op­po­site con­tri­bu­tions in the time turner sce­nario.

• you also need to draw a sur­face on the other side of the worm­hole.

Sure, and this will tell you that the Sun’s mass will not sud­denly ap­pear out of nowhere some place else, in ad­di­tion to not dis­ap­pear­ing from the So­lar sys­tem. An easy way to vi­su­al­ize the situ­a­tion is to think what hap­pens to the elec­tric field lines of an elec­tric charge trav­el­ing through a worm­hole.

• As an elec­tron ap­proaches a worm­hole, more and more of its elec­tri­cal field lines pen­e­trate the worm­hole, so its net charge ap­pears to drop on the lo­cal side and rise on the far side. The mass and elec­tri­cal charges do seem to ap­pear and dis­ap­pear out of nowhere. That’s why a worm­hole acts as a worm­hole in­stead of a solid wall.

Or is there some­thing go­ing on about worm­holes that I’m miss­ing?

• Or is there some­thing go­ing on about worm­holes that I’m miss­ing?

Yes, there is. It’s a com­mon er­ror made by those out­side of the sub­ject mat­ter, though. The sim­plest ar­gu­ment is that a worm­hole looks like a black hole from out­side, and any charge dropped into a black hole is con­served. More vi­su­ally, all the field lines re­main con­tin­u­ous be­tween the elec­tron and what­ever points in space the line goes through (oth­er­wise you’ll have to have lo­cal elec­tric fields sinks and sources, which can only be other charges), so the to­tal num­ber of the field lines per unit area (elec­tric field strength) re­main the same. Hence, the mea­sured charge re­mains the same, even long af­ter the origi­nal charge has gone through the worm­hole.

John Wheeler origi­nally in­vented this con­cept of “charge with­out charge” and “mass with­out mass” for other pur­poses.

So the worm­hole end in effect be­comes elec­tri­cally charged. What hap­pens at the other end is equally fas­ci­nat­ing.

• But that line of rea­son­ing as­sumes that a worm­hole has an event hori­zon. Aren’t worm­holes naked sin­gu­lar­i­ties?

• I get a feel­ing that you did not bother to read past the sec­ond line. The ar­gu­ment does not de­pend on the ex­is­tence of a hori­zon. And no, worm­holes are not naked sin­gu­lar­i­ties, they are not sin­gu­lar­i­ties at all, see­ing how one can get through un­mo­lested. I won­der if a Dis­cus­sion post on this topic, with pic­tures, more de­tails, and maybe some ac­tual calcu­la­tions would be of gen­eral enough in­ter­est.

• I read the first two sen­tences, and then you said ‘More vi­su­ally’, which sug­gests that it’s a restate­ment. It’s hard to think about dy­nam­ics in terms of field lines, and this is about a dy­nam­i­cal event, so I set it aside. Since that seems to be the heart of your ar­gu­ment, though, I’m fo­cus­ing on it now. And now it’s not clear to me that all of the usual iden­tities about elec­tri­cal field lines ap­ply in curved space.

Where does the no­tion of field lines come from? In the in­finite field lines limit, fol­low the elec­tri­cal field out­ward from each differ­en­tial an­gle of each charge. Note that you do this in­stan­ta­neously in some refer­ence frame. It’s not like the field lines trace a traversable path. What you’re trac­ing is offset at differ­ent times from the source charge. But if the charge is ca­pa­ble of not be­ing there at some time in the past, then there’s no prob­lem with that charge pro­vid­ing no field line—with that field line dis­ap­pear­ing with­out a charge pre­sent.

So in­stead, con­struct the charge’s field lines so that in­stead of be­ing at a pre­sent time, they are along the charge’s fu­ture light cone. You won’t find these field lines dis­ap­pear­ing in the mid­dle of nowhere. But you can use this to con­struct lon­gi­tu­di­nal elec­tro­mag­netic waves.

So some­thing is go­ing to break—ei­ther we get lon­gi­tu­di­nal elec­tro­mag­netic waves, or we get charges with­out charge. I don’t see that it’s clear that it has to be the sec­ond rather than the first. It de­pends what elec­tro­mag­netism IS.

(I seem to have mis­taken the defi­ni­tion of ‘naked sin­gu­lar­ity’, but ei­ther way there’s no event hori­zon.)

• It’s in­ter­est­ing how our in­tu­ition differ. After work­ing with curved space­time, sin­gu­lar­i­ties, black holes, worhholes and other ex­otic con­structs for a decade or so, I find it in­tu­itively ob­vi­ous that global charge con­ser­va­tion is not bro­ken by lo­cal effects, so no mat­ter how much weird­ness is go­ing on in­side some small vol­ume of space, the life must ap­pear nor­mal ev­ery­where else. In par­tic­u­lar, the usual Maxwell equa­tions, in­clud­ing the Gauss law must hold in the nearly flat nearly static space­time around this weird­ness, sim­ply by in­te­grat­ing the di­ver­gence of the elec­tric field over a closed sur­face en­com­pass­ing the re­gion in ques­tion. In fact, you can do a lit­tle bit of man­i­fold surgery to patch a worm­hole solu­tion to the flat space­time one, and then all your gar­den va­ri­ety EM ap­plies in the flat re­gion, with man­i­fest charge con­ser­va­tion and what not.

• Global charge con­ser­va­tion? I don’t see charge be­ing globally non-con­served. It just be­comes pos­si­ble to build sur­faces that look Gaus­sian but aren’t—they don’t com­pletely en­close a vol­ume.

Like, you have a worm­hole. Con­struct spheres A and B around each side of the worm­hole. To­gether, these con­sti­tute a Gaus­sian sur­face for a charge in­side them. Nei­ther one of them does, alone, be­cause there’s a way out through the worm­hole.

Mov­ing a charge from one side to the other, from a charge con­ser­va­tion point of view, isn’t differ­ent than if A and B were halves of a sin­gle sphere. A or B soaks up more or fewer of the elec­tric field lines.

• Like, you have a worm­hole. Con­struct spheres A and B around each side of the worm­hole. To­gether, these con­sti­tute a Gaus­sian sur­face for a charge in­side them. Nei­ther one of them does, alone, be­cause there’s a way out through the worm­hole.

...Still think­ing about the best way to show that the other half is ir­rele­vant. The stan­dard proof of the Stokes the­o­rem re­quires a com­pact man­i­fold with bound­ary, but the com­pact­ness con­di­tion does not hold ei­ther for a worm­hole, or for a black hole with a sin­gu­lar­ity. In the lat­ter case be­cause the sin­gu­lar­ity is not a part of the man­i­fold. The gen­er­al­iza­tion should be triv­ial, but evades me at the mo­ment.

• What does a charge halfway through a worm­hole look like?

Are these things one-way?

• Here is one link in­for­mally de­scribing the situ­a­tion:

If a pos­i­tive elec­tric charge Q passes through a worm­hole mouth, the elec­tric lines of force ra­di­at­ing away from the charge must thread through the aper­ture of the worm­hole. The net re­sult is that the en­trance worm­hole mouth has lines of force ra­di­at­ing away from it, and the exit worm­hole mouth has lines of force ra­di­at­ing to­ward it. In effect, the en­trance mouth has now been given a pos­i­tive elec­tric charge Q, and the exit mouth ac­quires a cor­re­spond­ing nega­tive charge -Q. Similarly, if a mass M passes through a worm­hole mouth, the en­trance mouth has its mass in­creased by M, and the exit mouth has its mass re­duced by an amount -M.

It doesn’t mat­ter if the worm­hole is one-way or not, but a one-way worm­hole ap­pears as a black hole on one end and a white hole on the other. You need some nega­tive en­ergy to sup­port a two-way worm­hole, which is, of course, a bit of a prob­lem to ob­tain, but it does not af­fect the above ar­gu­ment.

• So they’re say­ing that if you start with a charge far from a worm­hole on side A and drag it through to side B, then all of its elec­tri­cal field lines trace their way back through the worm­hole to side A again, just to main­tain con­ti­nu­ity of elec­tri­cal field lines?

Well, at least I’m sure I un­der­stand what they’re say­ing now. It doesn’t seem crazy any­more, but I’m not in the least con­vinced that it’s nec­es­sar­ily cor­rect.

• Step­ping back, I found what both­ers me here:

In par­tic­u­lar, the usual Maxwell equa­tions, in­clud­ing the Gauss law must hold in the nearly flat nearly static space­time around this weirdness

In line with lo­cal­ity, I see the deriva­tive forms of the Maxwell equa­tions as fun­da­men­tal, and the in­te­gral forms as a use­ful trick. When you go around chang­ing topol­ogy, these tricks no longer ap­ply naively.

The anal­ogy be­tween worm­hole and black hole is bro­ken. On the way across an event hori­zon, the charge gets in­finitely red-shifted, which im­mor­tal­izes it (slow­ing down an elec­tro­static charge doesn’t make it lack charge). Worm­holes, lack­ing an event hori­zon, lack this fea­ture as well.

Ba­si­cally, I just don’t see any sort of lo­cal mechanism for gen­er­at­ing this com­pen­sat­ing field. Do you have any­thing that doesn’t rely on a non­lo­cal/​in­te­gral for­ma­tion of Maxwell’s laws?

• OK, I’m pretty sure I have the ar­gu­ment.

First, I hope that we agree that what a dis­tant ob­server thinks the throat charge can be mea­sured by the field strength and cor­re­spond­ingly by the field flux through a Gaus­sian sur­face.

Now, if we take this Gaus­sian sur­face and shrink it around the charge into al­most a point, the flux through it will not change, if it does not cross other charges dur­ing shrink­ing. (That’s ba­si­cally the di­ver­gence the­o­rem for zero di­ver­gence.)

This shrink­ing is ob­vi­ously pos­si­ble when the charge is in some flat space far from the worm­hole throat. It re­mains pos­si­ble as the charge starts its trek into the throat. Since the space­time man­i­fold is con­tin­u­ous ev­ery­where, there are no ob­sta­cles to shrink­ing the Gaus­sian sur­face to noth­ing, no mat­ter how deep in­side the worm­hole the charge is. Even when it’s out the other side. All that hap­pens in that case is that your Gaus­sian sur­face has to travel through the throat and out (get­ting first smaller and then larger again in the pro­cess) be­fore fi­nally col­laps­ing onto the charge.

More for­mally, the con­di­tions for the Stokes the­o­rem hold re­gard­less of the man­i­fold’s topol­ogy, as long as the man­i­fold is con­tin­u­ous and differ­en­tiable. Speci­fi­cally, the man­i­fold with bound­ary re­quired by the the­o­rem, which is em­bed­ded in a spa­tial slice of the worm­hole space­time re­mains topolog­i­cally a solid sphere with the charge in­side it, no mat­ter where you move the charge.

Now, the ob­vi­ous ob­jec­tion is “if this is so, how come we don’t get the field of all other charges on the other side of the throat con­tribut­ing to the to­tal charge on this side?” And the an­swer is that they are not in­side the origi­nal Gaus­sian sur­face, i.e. you can­not shrink it to col­lapse on the point which was not origi­nally in it and did not move in­side dur­ing the sur­face col­lapse pro­cess.

This is all rather counter-in­tu­itive, given how imag­in­ing a curved 4D space­time takes some get­ting used to, but hope­fully it makes sense.

• First, I hope...

yup

This shrink­ing is ob­vi­ously pos­si­ble when the charge is in some flat space far from the worm­hole throat. It re­mains pos­si­ble as the charge starts its trek into the throat.

It is ap­prox­i­mately cor­rect when the charge is in some flat space far from the worm­hole throat and your not-quite-Gaus­sian sur­face also in­cludes that throat but doesn’t cut it off. The closer the charge gets to the throat, the worse this ap­prox­i­ma­tion is. That’s be­cause you’re go­ing to have some topolog­i­cal defect in your sur­face as it switches from con­tain­ing the worm­hole to not con­tain­ing the worm­hole. It’s not the tran­si­tion from one side to the other that’s the prob­lem. It’s the tran­si­tion from en­velop­ing the worm­hole to not en­velop­ing the worm­hole that’s the prob­lem.

If you never en­velop the worm­hole, then noth­ing you’re say­ing ever gets to ad­dress the ap­par­ent charge on the worm­hole.

More for­mally, the con­di­tions for the Stokes the­o­rem hold re­gard­less of the man­i­fold’s topol­ogy, as long as the man­i­fold is con­tin­u­ous and differ­en­tiable.

You’re cor­rect—the rule for Stokes the­o­rem, re­gard­less of the topol­ogy, is, you need to di­vide the space into an in­side and an out­side, and the in­side doesn’t get to ex­tend to in­finity. By draw­ing a sphere around one side of a worm­hole, you don’t ac­com­plish this.

• It is ap­prox­i­mately cor­rect when the charge is in some flat space far from the worm­hole throat and your not-quite-Gaus­sian sur­face also in­cludes that throat but doesn’t cut it off.

No, my point was that it is ex­actly cor­rect, not ap­prox­i­mately. The re­trac­tion ar­gu­ment is topolog­i­cal, not ge­o­met­ric, thus the cur­va­ture does not mat­ter. As long as the sur­face can be re­tracted all the way down to the charge, which it can, at least if you start some place far from the throat.

The closer the charge gets to the throat, the worse this ap­prox­i­ma­tion is. That’s be­cause you’re go­ing to have some topolog­i­cal defect in your sur­face as it switches from con­tain­ing the worm­hole to not con­tain­ing the worm­hole.

You don’t get any topolog­i­cal is­sues. There is not a sin­gle topolog­i­cal defect at any point which pre­vents the sur­face to re­tract onto the charge. This is en­sured by the con­ti­nu­ity of the man­i­fold.

you need to di­vide the space into an in­side and an out­side, and the in­side doesn’t get to ex­tend to in­finity. By draw­ing a sphere around one side of a worm­hole, you don’t ac­com­plish this.

As I said, since the re­trac­tion is never bro­ken, the charge always re­mains in­side the sur­face, even when it’s on the other side of the worm­hole.

Maybe I should make and post a pic­ture for a 2+1 case, now that I have it clear in my mind.

• There is not a sin­gle topolog­i­cal defect at any point which pre­vents the sur­face to re­tract onto the charge. This is en­sured by the con­ti­nu­ity of the man­i­fold.

Coffee cups are con­tin­u­ous too, but you can’t re­tract ar­bi­trary figures on them! That aside, you’re right that there’s no topolog­i­cal defect in the man­i­fold it­self. You can de­form a 2+1 pair of planes smoothly into a cylin­der. The topolog­i­cal prob­lem oc­curs in re­spect to the Gaus­sian sur­face. What used to be a cir­cle on the plane is turned by the pres­ence of the worm­hole into a cir­cle which goes around the cylin­der. IT NO LONGER HAS AN INSIDE.

• IT NO LONGER HAS AN INSIDE.

Yes it does. A con­tin­u­ous de­for­ma­tion does not change topol­ogy. I am now pretty sure that you are us­ing a wrong men­tal pic­ture. The Gaus­sian sur­face in ques­tion does not wind around the cylin­der, be­cause it never has. Think about it this way:

• we start with a very large cylin­der and a point on it

• we now draw a small cir­cle around this point.

• the cir­cle is con­tractible onto the point (by con­struc­tion)

• we now slide the charge along the cylin­der and ex­tend the sur­face around it to ac­com­mo­date, this does not af­fect the re­tractabil­ity in any way

• some part of the cylin­der can be pretty nar­row, cor­re­spond­ing to the worm­hole’s body, but it makes no differ­ence, the Gaus­sian sur­face will ex­tend through the worm­hole and back out, still en­com­pass­ing the charge in ques­tion, never once wind­ing around the cylin­der. It is sim­ply con­nected all the way through.

• I had ex­actly that men­tal pic­ture already, thanks. The prob­lem was that I was as­sum­ing that you were us­ing a Gaus­sian sur­face that ac­tu­ally helped es­tab­lish what you were try­ing to es­tab­lish.

See, what you said works so long as your origi­nal Gaus­sian sur­face did not con­tain the worm­hole.

If you never have a Gaus­sian sur­face con­tain­ing the worm­hole, then how the heck are you us­ing it as an ar­gu­ment con­cern­ing the charge of the worm­hole?

• OK, sorry for mak­ing an in­cor­rect as­sump­tion. It seems that I mi­s­un­der­stood your defi­ni­tion of “con­tain”. Ac­cord­ing to yours, in 3d the sphere around the worm­hole throat does not con­tain the throat, is this right? Given how it can slide through the throat and into the other side? If the worm­hole hap­pens to have the (spher­i­cal) event hori­zon, the event hori­zon does not “con­tain” the worm­hole throat? What you mean by con­tain­ing is that both throats must be in­side the sur­face, such that if we ex­cise the in­side of the sur­face and re­place it with a solid sphere, the whole of the worm­hole dis­ap­pears?

So, your ar­gu­ment is that, even through the Gaus­sian sur­face con­tains the charge, it does not con­tain the throat, and so some of the elec­tric field from the charge is free to es­cape to the other side, as the charge tra­verses the worm­hole? And that, once it is through, al­most all of the elec­tric field is on the other side?

• Your sec­ond para­graph sum­ma­riz­ing my po­si­tion is cor­rect. I don’t nec­es­sar­ily un­der­stand your first para­graph be­cause there are two senses of con­tain, but since you got the sec­ond para­graph right I trust that you meant the right things in the first.

(to clar­ify: the ‘con­tain’ in my above post was the naive sense of con­tain, like if the worm­hole is hid­den and you’ve got the re­gion of space it’s in sur­rounded—this sort of sur­face is still use­ful for gaug­ing the ap­par­ent charge on the worm­hole, but it’s not a true Gaus­sian sur­face)

So… what is wrong with this no­tion?

Well, be­fore you an­swer that, I’ll say what it oc­curs to me is wrong with it:

It feels differ­ent than other some­what analo­gous cases. In par­tic­u­lar, mag­netic fields sus­tained by su­per­con­duct­ing loops or plasma. Those mag­netic fields don’t just fade away. Once you thread the loop, it stays there.

On the other hand, the su­per­con­duct­ing loop is made of charges that con­tin­u­ously main­tain this field!

Once you’e look­ing at the ge­om­e­try of space, though, it’s not clear what’s go­ing to hap­pen. Are the fields con­tin­u­ously ra­di­ated by charges and masses, and if you do ge­om­e­try tricks to re­move the charges or masses the fields go away? Or are the fields things that can’t slip away like that?

Like… loosey-goosey imagery time! Con­sider a spher­i­cal shell of the elec­tri­cal field on a charge that’s 1 light sec­ond away from the charge. Is that a thing, or just a pat­tern?

If it’s a thing, then mov­ing the charge through the worm­hole won’t make it go through the worm­hole too. If it’s a pat­tern, then the charge mov­ing through the worm­hole makes the pat­tern go away.

• Hmm, static fields are not “ra­di­ated” by charges or masses, they ba­si­cally are charges or masses. That’s why you can­not tell a tiny worm­hole with a field threaded through it from a dipole, with­out look­ing closely down the throat.

I don’t think the anal­ogy with a pat­tern is a good one. Con­sider one elec­tric force line go­ing be­tween the charge and in­finity. As you move the charge, so does the line. But its two ends are firmly fixed at the charge and the in­finity cor­re­spond­ingly. As the charge goes through the worm­hole, this line is still “at­tached” to the in­finity out­side the throat, as it can­not just sud­denly dis­con­tin­u­ously jump be­tween the two asymp­totic re­gions. As a re­sult, the elec­tric field line gets “caught” in the topol­ogy, still go­ing back through the en­trance and out even af­ter the charge gen­er­at­ing it com­pletely tra­versed the worm­hole.

Here is a quote from an old pa­per http://​​arxiv.org/​​abs/​​hep-th/​​9308044:

It is also in­ter­est­ing to con­sider what hap­pens when a charged par­ti­cle tra­verses a worm­hole. (Of course, this “pointlike” charge might ac­tu­ally be one mouth of a smaller worm­hole.) Sup­pose that, ini­tially, the mouths of the worm­hole are un­charged (no elec­tric flux is trapped in the worm­hole). By fol­low­ing the elec­tric field lines, we see that af­ter an ob­ject with elec­tric charge Q tra­verses the worm­hole, the mouth where it en­tered the worm­hole car­ries charge Q, and the mouth where it ex­ited car­ries charge −Q. Thus, an elec­tric charge that passes through a worm­hole trans­fers charge to the worm­hole mouths.

• Static fields sure ACT like they’re ra­di­ated by charges - same causal struc­ture (see my re­cent post about causal­ity and static fields), same 1/​r^2. And of course we always think of the ra­di­a­tive fields as be­ing ra­di­ated by charges. So that cov­ers both cases. Based on what ac­tual lines of rea­son­ing do we not con­sider static fields to be ra­di­ated by charges?

With your ex­am­ple, why do we say the field lines are at­tached? It cer­tainly acts like it’s at­tached when­ever there are no worm­holes around and as long as charge is con­served, that’s for sure. But that might be a cause or it might be an effect. And when you dicky around with the as­sump­tions con­nect­ing them, it may or may not end up be­ing on the fun­da­men­tal side of things.

Like… for­get worm­holes for a mo­ment. Let’s go to a coun­ter­fac­tual—imag­ine there was a weak in­ter­ac­tion that vi­o­lated con­ser­va­tion of charge. As­sum­ing it ac­tu­ally hap­pened, against all ex­pec­ta­tions, what do you think the elec­tri­cal field would look like? If you trace it causally, what you see is un­usual but you have no trou­ble—it’s only in the spacelike cuts that it looks ugly. And physics re­ally re­ally doesn’t act like it’s im­ple­mented on spacelike cuts.

• Static fields sure ACT like they’re ra­di­ated by charges

I likely dis­agree with that, de­pend­ing on your mean­ing of “ra­di­ated”. I’d say they are “at­tached” to charges, acausally (i.e. not re­spect­ing the light cone). That’s what the static field ap­prox­i­ma­tion is all about.

Then there is the quasi-static case, where you ne­glect the ra­di­a­tion. The java ap­plet I linked shows what hap­pens there: the dis­tur­bance in the static field due to ac­cel­er­a­tion of charges prop­a­gates at the speed of light.

• Or is there some­thing go­ing on about worm­holes that I’m miss­ing?

Worm­holes of cos­mic pro­por­tions en­abling stars to move ac­cross—are just spec­u­la­tive ob­jects, do not for­get that!

• Yeah. I meant the­o­ret­i­cally. I wouldn’t even bother to ask if we were talk­ing about a real one, since the an­swer is ob­vi­ously go­ing to be ‘How would I know?’.

• If we were talk­ing about an ac­tu­al­ized worm­hole, the an­swer SHOULD be “Let’s find out!”

• … we’d love to, but it’s not as easy as it sounds.

• You should cross­post this to red­dit.com/​​r/​​hpmor

• Feel free, I don’t re­ally fol­low any par­tic­u­lar sub­red­dit.

• No men­tion of ap­parat­ing, AKA tele­por­ta­tion. If I had to solve this, I’d the­o­rize that both trans­mo­grifi­ca­tion and time turn­ers re­duce to the prob­lem of tele­por­ta­tion. Most other mag­ics can prob­a­bly also be ex­plained if you’re able to se­lec­tively tele­port things like forces, light, chem­i­cals, and var­i­ous forms of mo­men­tum or mo­men­tum-pro­duc­ing effects at a nanoscale near-in­stan­ta­neously (at least as far as the lo­cal mat­ter is con­cerned from the rele­vant refer­ence frames).

In other words, At­lantis con­tains a Big Badass Nan­otele­porter of Doom, or some­thing.

As for how that one would work: Beats me. When I try to work this deeply with rel­a­tivity, I’m gen­er­ally in way over my head and end up get­ting ev­ery sin­gle de­tail wrong in some way.

• Tele­por­ta­tion can work. A tar­geted short-lived neu­trino beam would do the trick, with a bit of work. At least you don’t have to break con­ser­va­tion of en­ergy or mo­men­tum.

• Tele­por­ta­tion can work. A tar­geted short-lived neu­trino beam would do the trick, with a bit of work.

Once you have tele­por­ta­tion you also have (the ap­pear­ance of) mass al­ter­ing trans­mo­grifi­ca­tion by sim­ple ap­pli­ca­tion of the former to mass un­known to the ob­servers. One you have ap­par­ent trans­mo­grifi­ca­tion and ad­vanced-pre­dic­tion-in-ad­vance you have Time Turn­ing. Again, through ap­ply­ing tele­por­ta­tion to care­fully syn­the­sized and se­lected mat­ter that the ob­servers aren’t aware of. For ex­am­ple, to a fully syn­the­sized ex­trap­o­la­tion of “MoR!Harry!2.hours.from.now”. From first per­sonal per­spec­tive of Harry this is equiv­a­lent to time travel.

This is childs play to Omega, it’s nearly ex­actly what he is con­trived for in New­comb’s Prob­lem. It may be a tad more difficult for a mere su­per­in­tel­li­gence but it re­mains an en­g­ineer­ing prob­lem, not so much a physics one.

At least you don’t have to break con­ser­va­tion of en­ergy or mo­men­tum.

The above pos­si­bile solu­tion isn’t “true” time travel. But it is to time travel pre­cisely what tar­geted short-lived neu­trino beams are to per­ceived “ap­par­i­tion”. There is lit­tle qual­i­ta­tive differ­ence in ei­ther re­la­tion­ship to de­scribed mag­i­cal events or the phisi­cal pos­si­bil­ity.

• It gets scarier if you re­al­ize that any­thing suffi­ciently ad­vanced to model “MoR!Harry!2.hours.from.now” can prob­a­bly model “MoR!Harry!2.hours.from.now+afraid.to.mess.with.time”.

• My head scratcher with tele­po­rata­tion is what would hap­pen to mo­men­tum? If a Nightcrawler (x-man tel­port­ing charater) type jumped off an air­plane and tele­ported mid-drop, would he still con­tinue mov­ing af­ter­wards? I’d as­sume so, but say he changed po­si­tion, tele­ported up­side down. Now what?

Ba­si­cally, is mo­men­tum rel­a­tive to an in­ter­nal or ex­ter­nal frame of refer­ence? I sus­pect the an­swer is, that isn’t a ques­tion be­cause the premises are im­pos­si­ble. Just like you can’t di­vide by zero, you can’t mul­ti­ply by im­pos­si­ble.

• En­ergy and mo­men­tum con­ser­va­tion is a lo­cal law in Rel­a­tivity. How such con­ser­va­tion laws-com­pli­ant Nightcrawler would land de­pends on the ac­tual mechanism of tele­por­ta­tion. For ex­am­ple, if he is trans­formed into a neu­trino beam for the du­ra­tion of the trans­port, the nec­es­sary mo­men­tum and an­gu­lar mo­men­tum can be im­parted by what­ever sur­rounds him at the points of de­par­ture and ar­rival. The air­plane or the sur­round­ing air will re­coil a bit (or a lot), and so will what­ever ob­jects he brakes against upon ar­rival.

• I don’t think mo­men­tum is con­served in Rel­a­tivity; how could it be, if mass and there­fore in­er­tia are not?

• En­ergy-mo­men­tum is most em­phat­i­cally con­served in rel­a­tivity, just not en­ergy and mo­men­tum sep­a­rately.

• What are the units of en­ergy-mo­men­tum? Mo­men­tum is mass­dis­place­ment/​time, while en­ergy is length^2mass/​time^2, so the con­ver­sion ra­tio would have to have units of length^2/​dis­place­ment-time. You can use some tricks to make it ap­pear to have units of length/​time (speed), but then you need to per­mit ei­ther mo­men­tum or en­ergy to be nega­tive.

Is the bind­ing en­ergy vari­able de­pend­ing on ve­loc­ity? Los­ing mass at rest does con­serve mo­men­tum, while los­ing mass in mo­tion does pro­por­tionately to the speed of the lost mass. (and no­tably, NOT de­pend­ing on whether it was mov­ing to­wards, away, or lat­eral to the ob­server- red/​blue shift is ir­rele­vant). Is the amount of en­ergy ob­served to be re­leased from a given atomic re­ac­tion vari­able de­pend­ing on the speed of the re­ac­tants?

• Maybe the per­son us­ing the Time Turner is just con­verted to air molecules of the same mass… and in the past, the molecules of air con­vert to the given per­son.

• Maybe the per­son us­ing the Time Turner is just con­verted to air molecules of the same mass… and in the past, the molecules of air con­vert to the given per­son.

So, peo­ple are about 700 times as dense as air- a 45 kilo girl will go from oc­cu­py­ing about .042 m^3 to about 30 m^3. If done slowly, this isn’t a prob­lem- but if you do the swap in­stan­ta­neously, you need to have all of those molecules in that small vol­ume. The pres­sure re­quired (i.e. the pres­sure that it will ex­pand out­ward with) with­out chang­ing the tem­per­a­ture is 700 atm; this is com­pa­rable to the max­i­mum cham­ber pres­sure of a firing pis­tol. The over­pres­sure from the Ok­la­homa City bomb­ing ex­plo­sion was only about 40% that large.

In the re­verse di­rec­tion, 30 m^3 (the size of a room that’s 12 feet by 11 feet by 8 feet, or a sphere with a 3.1m ra­dius) of air will be re­quired. If done too quickly, this could cause similar prob­lems.

• To sum­ma­rize what Vaniver said: This is ex­actly equiv­a­lent to the per­son ex­plod­ing into bits, ex­cept with even more pres­sure and the bits are solid com­pressed air fly­ing around like shrap­nel and kil­ling ev­ery­one nearby with sheer pres­sure.

• The time turn­ers don’t work as they are de­scribed; can you make them work? As­sume you’re op­er­at­ing in HP:MoR-verse, if it helps.

(Thank you for the treat.)

• can you make them work?

Without break­ing some of our laws of physics? Then you’d just be ask­ing shminux if he can build a re­verse-time ma­chine (go­ing for­ward in time is easy ;-).

As­sume you’re op­er­at­ing in HP:MoR-verse

Ok: magic!

• I tried in this re­ply. Not sure if you find it to your satis­fac­tion.

• Now an­other, de­cep­tively similar ques­tion: if the Sun dis­ap­pears this in­stant, how long be­fore the Earth will stop or­bit­ing the point where it used to be? The com­mon an­swer: grav­ity trav­els with the speed of light, so also 8.5 min. This an­swer is ob­vi­ous, sim­ple and wrong. Yes, dead wrong. Why? be­cause static grav­ity is not like light, it’s more like elec­tric field, only worse.

I think you’re call­ing this more wrong than you should be. You fol­low it up by ar­gu­ing that if the sun in­stantly dis­ap­pears, there’ll never be a grav­ity change, so the 8.5min is wrong. But re­ally, what you’re do­ing is show­ing that the sun can’t in­stantly dis­ap­pear in the sense that peo­ple mean the ques­tion. The nat­u­ral fol­low-up is to mod­ify the ques­tion to some­thing like the “what if the sun split into two halves that rapidly moved apart” ver­sion, and in that case, un­less I’m mis­taken, it would be 8.5 min­utes be­fore the earth’s or­bit changed.

• Nice ap­pli­ca­tion of the point of Univer­sal Fire to HPMoR. (Eliezer gets a pass on this, IMO, be­cause it’s some­one else’s fic­tional uni­verse he’s work­ing in.)

• I’m dis­ap­pointed in this thread—not be­cause there are mis­takes here but be­cause posters are not com­plet­ing the due dili­gence of ad­mit­ting their own ig­no­rance on the sub­ject. Gen­eral rel­a­tivity is known to not be sim­ple and some claims made here wouldn’t even pass muster in the much sim­pler New­to­nian world. We should all be prac­tic­ing the skill of say­ing ‘this sub­ject is deemed com­plex and so I am quite likely to be mak­ing a mis­take given that I haven’t stud­ied it in depth.’ Usu­ally at this point we have some grad stu­dents study­ing the sub­ject step­ping in to clear up mis­con­cep­tions, but I don’t see any here so I’m speak­ing up even though I have no train­ing in GR. (And to take my own ad­vice: I could be wrong about this all! please let me know if I am. But call­ing out an ar­gu­ment takes less than mak­ing one so my job here is eas­ier.)

Okay, so for some speci­fics: How about the over-ap­pli­ca­tion of Gauss’s Law as a tool to say “it doesn’t mat­ter what hap­pens, the net effect is the same on Earth’. Gauss’s Law only says that about spher­i­cally sym­met­ric mass dis­tri­bu­tions and quite a few of the situ­a­tions here are not that. (Other­wise the three body prob­lem would be triv­ial to solve!) Or how about some of the com­ments here that in­volve ac­cel­er­at­ing mas­sive bod­ies to near light-speed very rapidly. Sure this is fine in New­to­nian me­chan­ics, but in GR the ac­cel­er­a­tion of a large body cre­ates grav­i­ta­tional waves, etc. It’s not so sim­ple as it’s be­ing made to seem!

I ex­pect more dili­gence and cau­tion from a com­mu­nity that val­ues be­ing cor­rect over sound­ing cor­rect! And here I will ap­plaud those who has shown un­cer­tainty in this, such as RichardKennedy.

(Note: I wasn’t one of the ones who origi­nally down­voted shmin­uxl.)

• Upvoted for first para­graph. Nearly took it back for con­fi­dently as­sert­ing false state­ments about Gauss’s Law, in di­rect con­tra­dic­tion of point in first para­graph.

• Please note the un­con­fi­dence already stated in the first para­graph and the point that I should ex­pect lower stan­dards for my­self to be able to say “hey I don’t think this is right” ver­sus “here’s a strong ar­gu­ment I cre­ated whole-cloth.” Se­condly, I am still rea­son­ably con­fi­dent about the state­ments about Gauss’s law that I tried to make, though there is a good chance I have mis­com­mu­ni­cated them. See my re­ply to shminux for an at­tempt at clar­ifi­ca­tion.

• As an ex­pert in the area:

Gauss’s Law only says that about spher­i­cally sym­met­ric mass dis­tri­bu­tions and quite a few of the situ­a­tions here are not that.

False, it ap­plies to any closed sur­face. Spher­i­cal sym­me­try is just the sim­plest prob­lem where the Gauss law can be use­fully ap­plied.

(Other­wise the three body prob­lem would be triv­ial to solve!)

The three-body prob­lem has no re­la­tion to the Gauss law, ex­cept in­so­far as it is equiv­a­lent to the in­verse square law for point masses in flat space.

Or how about some of the com­ments here that in­volve ac­cel­er­at­ing mas­sive bod­ies to near light-speed very rapidly. Sure this is fine in New­to­nian me­chan­ics, but in GR the ac­cel­er­a­tion of a large body cre­ates grav­i­ta­tional waves

This is true in most cases, but not in spher­i­cal sym­me­try (there are no spher­i­cally sym­met­ric grav­i­ta­tional waves). That’s why I could write “an amount of mat­ter equiv­a­lent to her mass has to travel in­wards out of nowhere and co­a­lesce into a per­son” and not worry about en­ergy losses to grav­i­ta­tional ra­di­a­tion. Though there would be, of course, a lot of losses to EM ra­di­a­tion if the mat­ter in ques­tion con­sisted of charged par­ti­cles.

Hope this makes sense.

• False, it ap­plies to any closed sur­face. Spher­i­cal sym­me­try is just the sim­plest prob­lem where the Gauss law can be use­fully ap­plied.

Oh I ab­solutely don’t deny that. What I do as­sert is that you can’t ig­nore what hap­pens on the in­side of a closed sur­face if you don’t know that the mass is spher­i­cally sym­met­ric and you want to calcu­late the force at a spe­cific force. If this is not what you are say­ing, then this is a mis­com­mu­ni­ca­tion not a dis­agree­ment. What I took you to mean when you made com­ments like “This still con­serves mass, since black holes have mass. It also obeys the Gauss Law” is to mean not just that Gauss’s law holds, but that it’s in­te­gral form can still be used to calcu­late the force on the Earth in a triv­ial mat­ter de­pend­ing only upon the mass in­side the closed sur­face and hence then the force on the Earth would stay the same as if the sun had not split in two.

I hope that we both agree that Gauss’s law could not be used in such a man­ner (as it only gives the in­te­gral of flux over the sur­face and so with­out sym­me­try this force is not con­stant over the sur­face). It be­lieve that you would also agree that split­ting the sun in two and send­ing them in op­po­site di­rec­tions as de­scribed would re­sult (in the case of New­to­nian grav­ity) in a de­crease in the grav­i­ta­tional force the Earth ex­pe­rienced. Given this, I am not sure how to char­i­ta­bly in­ter­pret the com­ment I quoted. Could you elab­o­rate on what you meant if you didn’t mean what I thought you meant? I will at this point cer­tainly ad­mit and apol­o­gize for the fact that I in­ter­preted your com­ments un­char­i­ta­bly since it sounded like you were mak­ing a com­mon be­gin­ner’s er­ror.

My three-body com­ment meant this: if we could use Gauss’s law to say that the force from in­side of an en­closed area is di­rectly pro­por­tional to the mass in­side, then we would be able to draw a re­gion in space around two of the three bod­ies and calcu­late the force on the third. This force would then de­pend only upon the to­tal mass which is con­stant and would point to the cen­ter of the en­closed area. So then we’d be able to solve for that one body’s mo­tion in­de­pen­dent of the other two. This is patently ab­surd and not pos­si­ble. I chose to give it as an ex­am­ple of how such a naive ap­pli­ca­tion of Gauss’ law would give ob­vi­ously in­cor­rect re­sults but the illu­sion of trans­parency grabbed me and I failed to make my­self clear.

This is true in most cases, but not in spher­i­cal sym­me­try (there are no spher­i­cally sym­met­ric grav­i­ta­tional waves).

That com­ment was not di­rected at your origi­nal post but at some of the com­ments on split­ting the sun in two and shoot­ing them apart. Sorry for the con­fu­sion!

• What I took you to mean when you made com­ments like “This still con­serves mass, since black holes have mass. It also obeys the Gauss Law” is to mean not just that Gauss’s law holds, but that it’s in­te­gral form can still be used to calcu­late the force on the Earth in a triv­ial mat­ter de­pend­ing only upon the mass in­side the closed sur­face and hence then the force on the Earth would stay the same as if the sun had not split in two.

This would be a novice mis­take I have been cor­rect­ing countless times as a tu­tor and TA. The Gauss law holds even when there is no sym­me­try, but it is much less use­ful to calcu­lat­ing elec­tric or grav­i­ta­tional field at a given point. It is, how­ever, can be prof­itably used to ar­gue other points, like the one I made.

I will at this point cer­tainly ad­mit and apol­o­gize for the fact that I in­ter­preted your com­ments un­char­i­ta­bly since it sounded like you were mak­ing a com­mon be­gin­ner’s er­ror.

No need for an apol­ogy, I’m glad we cleared that up.

if we could use Gauss’s law to say that the force from in­side of an en­closed area is di­rectly pro­por­tional to the mass in­side, then we would be able to draw a re­gion in space around two of the three bod­ies and calcu­late the force on the third.

I un­der­stand what you mean now, given what you said pre­vi­ously and I cer­tainly agree that this would make no sense.

In­fer­en­tial dis­tance is a b****.

• What if we turn the suns mass into two lasers that shot out equally from the poles?

• Maybe we could do that. But the laser pulses short and nar­row and there­fore dense enough are two black holes made of pure light. Two black holes, leav­ing the scene with the ex­act speed of light.

Not much differ­ent from the already men­tioned solu­tion, albeit even cooler.

• Two black holes, leav­ing the scene with the ex­act speed of light.

Black holes have rest mass, made of light or not, so this is im­pos­si­ble. Can you spot the er­ror?

Hint: one pho­ton has no rest mass, while a bunch of pho­tons in a box (pho­ton gas) does.

• Black holes have rest mass, made of light or not, so this is im­pos­si­ble. Can you spot the er­ror?

It is true that black holes have rest mass, re­gard­less of the form of mass-en­ergy used to cre­ate them. As such it is im­pos­si­ble for them to travel at the speed of light. How­ever the er­ror-hint you give is mis­lead­ing.

one pho­ton has no rest mass, while a bunch of pho­tons in a box (pho­ton gas) does.

This is true but in­napli­ca­ble. Mak­ing a black hole out of pho­tons re­quires that the pho­tons are not all mov­ing in the same di­rec­tion. For an ex­pla­na­tion see here. Alter­nately, sim­ply con­sider the case where all the en­ergy is con­cen­trated into a sin­gle (ridicu­lously) high en­ergy pho­ton. As you ob­serve, “one pho­ton has no rest mass”. No black hole is formed by a sin­gle (un­con­fined) pho­ton. The ex­pla­na­tion is some­what similar to why you don’t form a black hole if you travel re­ally fast.

• There is no er­ror here.

http://​​en.wikipe­dia.org/​​wiki/​​Kugelblitz_%28as­tro­physics%29

You CAN have a black hole made of only pho­tons. Of neu­trinos even.

• There is no er­ror here.

I ten­ta­tively sug­gest that there may be an er­ror in a de­tail here, albeit not the one shminux sug­gests.

http://​​en.wikipe­dia.org/​​wiki/​​Kugelblitz_%28as­tro­physics%29

You CAN have a black hole made of only pho­tons. Of neu­trinos even.

You can in­deed (and such black holes are even more cool). How­ever I be­lieve the for­ma­tion of a black hole from pho­tons re­quires that the pho­tons aren’t all pro­pogat­ing in the same di­rec­tion. This is a good thing be­cause if you did have a black hole trav­el­ing at the ex­act speed of light then, well, re­al­ity is all bro­ken down. Black holes have rest mass; speed of light travel is off limits. In gen­eral it doesn’t re­ally mat­ter (so to speak) whether the mass-en­ergy that they are cre­ated from is pho­tons or a mix of pa­per­clips and left over Babyeater offspring.

• Of course you can. But they can­not travel at light speed.

• The so called Kugelblitz can of course travel with the speed of light.

www.en.wikipe­dia.org/​​wiki/​​Kugelblitz_(as­tro­physics)

Imag­ine a laser beam pulse, 1 nanosec­ond long. A foot long and an inch tick cylin­der of light is trav­el­ing with the speed of light, into the dark­ness of space. When it passes by an atom, there is a brief grav­ity effects be­tween the two. The atom and the beam.

Now imag­ine two par­allel beams, 1 kilo­me­ter long. The grav­ity effect is even larger. You can pile as many laser beams to­gether, as you wish. You can make them (light) years long. Even­tu­ally, the pass­ing atom can’t es­cape the beam’s grav­ity field, for the es­cape ve­loc­ity is greater than c. The beam is a black hole.

Trav­el­ing with the speed of light!

• You can fix the bro­ken link by es­cap­ing the first clos­ing paren­the­sis: `[a](http://​​some­where.com/​​some­thing(stuff\))`

• In that case it didn’t form a black hole un­til it in­ter­acted with the atom, and at that mo­ment it slowed down from light speed. How do I know?

Un­til the atom came along, you’re free to use any old refer­ence frame with­out com­pli­cat­ing mat­ters ter­ribly. So you can ar­bi­trar­ily red-shift that laser pulse to the point that the en­ergy den­sity is triv­ial.

If you do that, you shift en­ergy into the atom. So, it’s the col­li­sion that al­lows it to con­dense into a black hole.

• It is rather in­ter­est­ing ques­tion, what hap­pens when the pure light black hole in­ter­acts with an atom. I have no idea, ex­cept that is in­ter­est­ing to con­sider this kind of ques­tions. What hap­pens un­der some pe­cu­liar cir­cum­stances. Do the cur­rent the­o­ries work there or break down?

I don’t know the an­swer, who does?

• As I men­tioned be­fore, there is no such thing as a “pure light black hole”. All un­charged black holes are pure vac­uum, re­gard­less of how they formed. There are also no beam-shaped black holes (all black holes are spher­i­cal in shape, this is a well known re­sult in Gen­eral Rel­a­tivity), though a reg­u­lar black hole can the­o­ret­i­cally form from light alone (kugelblitz) un­der cer­tain rare cir­cum­stances.

Your model has a num­ber of tech­ni­cal er­rors which pre­vent it from work­ing. For ex­am­ple, you can­not form a black hole by shoot­ing a light pulse from a laser, ex­cept maybe by fo­cus­ing it, in which case the cen­ter-of-mass ve­loc­ity will be sub­light and you end up ei­ther cre­at­ing mul­ti­ple black holes or feed­ing a sin­gle one, de­pend­ing on the de­tails. More­over, while light does curve space­time, it’s not as sim­ple as the grav­i­ta­tional at­trac­tion be­tween mas­sive bod­ies. There are many known solu­tions which in­clude perfect null fluid (that’s what a con­tin­u­ous beam of light is), none are triv­ial.

If you still think that you can think up some­thing that 100 years of GR re­search by the ge­niuses like Ein­stein, Hawk­ing and oth­ers did not no­tice, con­sider learn­ing the sub­ject se­ri­ously first.

• I haven’t in­vented a thing in Gen­eral Rel­a­tivity, black holes and so on. I have no in­ten­tion.

We all are just talk­ing and link­ing here. You have started this thread and you ob­vously dis­like some com­ments.

• Where does the mo­men­tum come from to ac­cel­er­ate the atom? From deflect­ing the pho­tons, which then di­verge?

• That cer­tainly works, in the­ory, though to emit the sun’s mass as light in a short enough time, you will have to out­shine a quasar by or­ders of mag­ni­tude (try a Fermi es­ti­mate on this). It will prob­a­bly heat up the dust around the Sun enough to bake and evap­o­rate all of the So­lar sys­tem in its heat and scat­tered laser light. Also note that this would not con­tra­dict my as­ser­tion that stuff has to travel out­ward, not just dis­ap­pear.

• It could be that she turns into neu­trinos, which would go un­no­ticed, or tachyons, al­low­ing her to phys­i­cally travel back in time, or both if a cer­tain ex­pla­na­tion of the neu­trino anomaly is to be be­lieved.

• The OPERA anomaly was due to… well...

• Who down­votes just be­cause some­one hasn’t heard the lat­est news about the OPERA anomaly? Upvoted to neu­tral.

• Don’t sweat the vote. In the last few weeks I get flash-down­vote spells (5-10 points drop in a mat­ter of min­utes on un­re­lated old com­ments) nearly ev­ery day, pre­sum­ably from some anony­mous cow­ard(s) who are quietly pissed at me, and Mer­lin knows I said enough on this fo­rum to piss off some oth­er­wise ra­tio­nal peo­ple. Others com­plained of the same treat­ment be­fore, hence no down­vote but­ton on the user page, but there is lit­tle one can do against a de­ter­mined at­tacker.

Any­way. I find it best to stay away from karma dis­cus­sions other than ask­ing for feed­back on un­usu­ally high up­votes or down­votes.

• So what hap­pens af­ter the Vo­gons have finished suck­ing the mass through their worm­hole and close it off? Does the earth end up or­bit­ing a patch of empty space that still ex­erts grav­ity, or was that sec­tion ac­tu­ally an ar­gu­ment that they can’t open/​close a worm­hole and suck the sun through it?

• It will not be just empty space, it will be a black hole as mas­sive as the Sun was. Rather counter-in­tu­itive, I know. But note that black hole is “a patch of empty space that [...] ex­erts grav­ity”.

• But if you give up on Gen­eral Rel­a­tivity, quite a few things will un­ravel, like all four New­ton’s laws.

“Weasley says that rock­ers use a spe­cial kind of sci­ence called op­po­site re­ac­tion, so the plan is to de­velop a jinx which will pre­vent that sci­ence from work­ing around Azk­a­ban.”

• mass can­not just dis­ap­pear, it has to spread out.

This is not en­tirely true. At least in prin­ci­ple, the Sun can be di­vided into two black holes. One con­sists the Sun’s north half, the other one of the Sun’s south half, each go­ing in the op­po­site di­rec­tion with the nearly light speed, per­pen­dicu­lar to the eclip­tic plane.

Earth would feel a rapidly fad­ing grav­i­ta­tional pull and the dark­ness.

The ques­tion is, can we do it even bet­ter? Can the Sun dis­ap­pear in the op­po­site di­rec­tion, away from us with the (nearly) speed of light, with­out spray­ing us with some deadly rays?

Yes, it can. Can any­body figure it out how?

• each go­ing in the op­po­site di­rec­tion with the nearly light speed, per­pen­dicu­lar to the eclip­tic plane.

This still con­serves mass, since black holes have mass. It also obeys the Gauss Law

• Who’s down­vot­ing this? It’s ob­vi­ously cor­rect, while the post it’s re­spond­ing to is com­pletely wrong on its first, most im­por­tant state­ment.

• Would this work?

• It would work, to a de­gree. But for the ma­jor­ity of the Sun’s mass to es­cape from us with the nearly light speed, it wouldn’t. The small pieces should be very close to the speed of light and thus very heavy for an ob­server here on Earth. That would mean an even big­ger grav­i­ta­tional pull to­ward the ex-Sun, for a while.

• The ques­tion is, can we do it even bet­ter? Can the Sun dis­ap­pear in the op­po­site di­rec­tion, away from us with the (nearly) speed of light, with­out spray­ing us with some deadly rays?

Sure, it just has to make sure it’s rays (or rel­a­tivis­ti­cally ac­cel­er­ated mass pack­ets) miss us. Shoot­ing two (or more) si­mul­ta­neous streams of mass on ap­pro­pri­ate vec­tors that add up to di­rectly away from us should work fine… or it would in a true vac­uum. Depend­ing on the de­tails of the im­ple­men­ta­tion I’d be a lit­tle wor­ried about in­di­rect rays hit­ting earth due to col­li­sions with dust.

Yes, it can. Can any­body figure it out how?

What is your solu­tion? The ob­vi­ous one (above) re­quires the Sun sac­ri­fic­ing rather a lot of mass in the pro­cess of run­ning away. Does your solu­tion avoid this?

• I have a solu­tion, which by its effect, is nearly in­dis­t­in­guish­able from the “in­stantly dis­ap­pear­ing Sun”.

You have to con­vert all the Sun’s mass into a Dyson sphere at the Mer­cury’s or­bit. For the Sun’s now miss­ing light, it can be a hot spot some­where on the Dyson. Ar­bi­trary perfect imi­ta­tion of the Sun shin­ning on the sky for us.

Now, the sphere has started to in­flate at the rate of a few km per sec­ond. Our or­bit would be just the same un­til the mo­ment the Dyson sphere reaches us. At the last few mil­lion kilo­me­ters the sphere’s in­fla­tion ac­cel­er­ates and we are en­veloped in­side oth­er­wise empty sphere in a mat­ter of sec­onds. We are in zero grav­ity of a hol­low sphere, shortly af­ter ev­ery­thing still seemed nor­mal. Of course, there is a hole in the sur­face of the DS just big enough that no col­li­sion takes place.

• We are in zero grav­ity of a hol­low sphere

Oh, that’s what the grav­ity from a hol­low sphere all adds/​mul­ti­plies out to? Uniform zero (net) grav­ity in­side, nor­mal out­side the sphere? Neat.

• An aside for those cu­ri­ous about the Gauss Law ar­gu­ment. The law in its in­te­gral form states that the flux of the grav­i­ta­tional field in­ward through any closed sur­face en­com­pass­ing the Sun is pro­por­tional to the Sun’s mass.

As long as the mass dis­tri­bu­tion is spher­i­cally sym­met­ric the grav­ity out­side of the sun is the same as if the mass was all lo­cated at the cen­ter. It’s the same for elec­tro­static force since that goes like 1/​r^2 too :D.

Oh, that’s what the grav­ity from a hol­low sphere all adds/​mul­ti­plies out to? Uniform zero (net) grav­ity in­side, nor­mal out­side the sphere? Neat.

• Yep! New­ton had a proof in Prin­cipia, but here’s a more re­cent, ge­o­met­ric one.

• At the cen­ter there is no net, but in­side the sphere the net effect is the same as if all the mass were con­cen­trated at the cen­ter. (Sup­pose that there was a nor­mal net effect ev­ery­where at and just above the sur­face and zero net effect just be­low the sur­face: that would re­quire that the grav­i­ta­tional field not be con­tin­u­ous, which is not the case)

• At the cen­ter there is no net, but in­side the sphere the net effect is the same as if all the mass were con­cen­trated at the cen­ter.

Re­mem­ber that this is a spher­i­cal shell, not a sphere. When calcu­lat­ing the force of grav­ity in­side the Earth, for ex­am­ple, you ig­nore all mass at higher ra­dius that the lo­ca­tion you’re in­ter­ested in, but not the mass at lower ra­dius.

Why does this work? Imag­ine op­po­site cones that origi­nate at the cen­ter of the spher­i­cal shell. The in­ter­sec­tion of that cone with the shell will have sur­face area that in­creases with r^2, but the in­verse square law de­creases with r^2, and so the grav­i­ta­tional effect only de­pends on the an­gle of the cone, re­gard­less of dis­tance- but the two cones are pointed in op­po­site di­rec­tions, and so can­cel out. (This is ob­vi­ous at the cen­ter, but works just as well el­se­where in­side the spher­i­cal shell.)

that would re­quire that the grav­i­ta­tional field not be con­tin­u­ous, which is not the case

For an in­finitely thin shell, the drop in grav­i­ta­tion is in­finitely steep. But shells with real thick­ness will have a grad­ual dropoff that cor­re­sponds to that thick­ness.

• … So pres­sure in a gaseous ce­les­tial body doesn’t in­crease lin­early; delta-p falls off to zero in the cen­ter?

• It’s not clear to me where that model came from. We shouldn’t ex­pect the den­sity of ce­les­tial bod­ies to be uniform un­less they’re made of some­thing in­com­press­ible, and it’s im­por­tant to sep­a­rate out the net force of grav­ity and the weight of mass above you. In steady state, all of the mass in the ce­les­tial body is be­ing pul­led to­wards the cen­ter by grav­ity, pushed up­wards by the mass be­low it, and has to push up­wards on the mass above it (with the net force be­ing 0). I haven’t done the math to see what the pres­sure func­tion would look like for a gaseous ce­les­tial body, and it seems like the full calcu­la­tion will have lots of com­pli­ca­tions, but we can note that the mass be­low you has to push up harder than the mass above you is push­ing down, sug­gest­ing the pres­sure is high­est at the cen­ter.

• delta-P is the first deriva­tive of pres­sure; it would have to be zero at the cen­ter for there to be a pres­sure max­i­mum at zero.

I would ex­pect a gaseous body to have a roughly spher­i­cally sym­met­ric mass dis­tri­bu­tion, which is all we need. Treat it as an in­finite num­ber of in­finitely thin spheres each of uniform den­sity, and we can do calcu­lus on it.

We can also do this though ex­per­i­ment with a perfect liquid of uniform den­sity; at least it will have a sur­face that we can stop at. Pres­sure is still high­est at the cen­ter and re­al­ity is con­tin­u­ous, mean­ing dP/​dR is zero at the cen­ter and ap­proaches zero as R ap­proaches zero.

Sur­face grav­ity of a sphere of con­stant den­sity and ra­dius R is pro­por­tional to R? Mass is pro­por­tional to vol­ume (R^3) and sur­face grav­ity is pro­por­tional to mass/​R^2, or R^3/​R^2, or R.

Okay, I’ve got a new re­spect for the prob­lems in­volved with us­ing baro­met­ric pres­sure to mea­sure al­ti­tude, and the ad­van­tages of us­ing baro­met­ric pres­sure di­rectly for nav­i­ga­tional pur­poses at high al­ti­tudes.

• Can you please use ques­tions in­stead of con­fi­dent con­tra­dic­tion when you’re up against New­ton?

• At the time of writ­ing, I thought I was para­phras­ing New­ton. Down­branch I re­al­ized that the be­hav­ior of be­ing at­tracted to the cen­ter of a shell while in­side the shell would, in a body com­posed en­tirely of fluid, yield a max­i­mum first deriva­tive of pres­sure at the cen­ter, and no max­i­mum of pres­sure within the body.

• This will in­deed work, but it has no re­la­tion to your origi­nal puz­zle:

Can the Sun dis­ap­pear in the op­po­site di­rec­tion, away from us with the (nearly) speed of light, with­out spray­ing us with some deadly rays?

• It is your fre­quent re­mark: “What has this to do with what has been pre­vi­ously said?”

;)

• Why bother with black holes?

• Be cause they are cool. They don’t shine very bright. As long as they en­cap­su­late all the mat­ter and don’t have any ac­cre­tion discs.

Other­wise you have to cool down ev­ery­thing for many mil­lion Kelv­ins.

• Why bother with black holes?

Bet­ter ques­tion: why not bother with black holes? I mean, since we’re talk­ing about the im­pli­ca­tions do­ing crazy im­pos­si­ble stuff to re­al­ity any­way...

Mind you, I can’t find any refer­ence or al­lu­sion to black holes in the grand­par­ent so I’m not en­tirely sure why you are ask­ing me.

• I sus­pect that some­one is sys­tem­at­i­cally down­vot­ing ev­ery com­ment made to this post, but what the heck: it’s in­ter­est­ing, I’ll add my two cents and gladly take the down­vote. ’gim­mie your best shot pal, I can take it.
About a pos­si­ble way for a time-turner to work...
IIRC, in or­der to make an Alcu­bierre wave more effi­cient, a space-time met­ric was pro­posed that in a sense iso­lated the ship from the out­side uni­verse, ex­pos­ing only a very tiny sur­face. This al­lowed the wave to carry away the ob­ject with­out re­quiring the en­ergy of all stars in our galaxy.
I can as­sume that Time-Turn­ers, since they can work only hour-by-hour, de­ploy a very tiny wor­mo­hole ev­ery hour (!!) in a con­nected chain, and when the user ac­ti­vate it, it en­closes him/​her in this bub­ble and trans­port him/​her through one of the point of the chain.
Of course I can­not calcu­late it, but I sus­pect that the prob­lem lies in the bub­ble for­ma­tion: I don’t know if the en­ergy spent to form it can only be di­rected in­wards or through the holes...
But hey, it’s magic!

• I sus­pect that some­one is sys­tem­at­i­cally down­vot­ing ev­ery com­ment made to this post, but what the heck: it’s in­ter­est­ing, I’ll add my two cents and gladly take the down­vote. ’gim­mie your best shot pal, I can take it.

It is clearly not the case that some­one is sytem­at­i­cally down­vot­ing ev­ery com­ment made to this post. It is a pre­dic­tion that is eas­ily tested by hov­er­ing the mouse poin­ter over the karma of the com­ments. This will show you that many com­ments, in­clud­ing some of the older ones, are 100% pos­i­tive and many more are 0% pos­i­tive with 0 karma. Both of those states pre­clude them hav­ing been down­voted, sys­tem­at­i­cally or oth­er­wise.

I am about to down­vote this com­ment with­out read­ing fur­ther. I en­courage oth­ers to do like­wise (un­less you edit out the bluster). As a rule of thumb nearly any com­ment that con­veys “I’m go­ing to get down­voted but ” is worth downt­ing.

• It is clearly not the case that some­one is sytem­at­i­cally down­vot­ing ev­ery com­ment made to this post. It is a pre­dic­tion that is eas­ily tested by hov­er­ing the mouse poin­ter over the karma of the com­ments. This will show you that many com­ments, in­clud­ing some of the older ones, are 100% pos­i­tive and many more are 0% pos­i­tive with 0 karma. Both of those states pre­clude them hav­ing been down­voted, sys­tem­at­i­cally or oth­er­wise.

Ah, I didn’t knew about that func­tion: in that case, your anal­y­sis is ab­solutely cor­rect! I re­tract my state­ment.

That, on the other side, seems wrong. You should down­vote and read for­ward.

I en­courage oth­ers to do like­wise (un­less you edit out the bluster)

That seems very wrong: in an en­vi­ron­ment that pred­i­cates ra­tio­nal­ity, stat­ing that you’re wrong should be en­couraged, not pun­ished. It’s not about the down­vot­ing, as I’ve writ­ten, it’s about down­vot­ing un­less I can­cel the origi­nal state­ment. The shift seems to have moved, at least in your com­ment, from be­ing less wrong to be­ing right.

I re­fuse such a drift. I won’t edit out the state­ment (but I’ll gladly ad­mit I was wrong).

• That seems very wrong: in an en­vi­ron­ment that pred­i­cates ra­tio­nal­ity, stat­ing that you’re wrong should be en­couraged, not pun­ished. It’s not about the down­vot­ing, as I’ve writ­ten, it’s about down­vot­ing un­less I can­cel the origi­nal state­ment.

Par­don me, I wasn’t clear. Re­trac­tion and (noted, trans­par­ent) edit­ing can some­times be even bet­ter than re­moval so of course can be in­cluded in the ex­cep­tion to the sug­ges­tion.

The shift seems to have moved, at least in your com­ment, from be­ing less wrong to be­ing right.

Note that wrong­ness isn’t the is­sue. It is the com­bi­na­tion “need­less provo­ca­tion && wrong”. For ex­am­ple if there wasn’t available proof that the claim was wrong then I would per­haps still have down­voted just be­cause the karma-drama is an­noy­ing but I would be un­likely to have also made a com­ment ex­press­ing crit­i­cism of the be­hav­ior.

• I re­tract my state­ment.

You might want to edit your com­ment if you want peo­ple to stop down­vot­ing.

• As a rule of thumb nearly any com­ment that con­veys “I’m go­ing to get down­voted but ” is worth downt­ing.

I agree, oth­er­wise it’ an empty challenge :)