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 :)