# Information loss in theoretical physics **URL:** \ **Category:** Factual Questions\ **Created:** [December 23, 2021, 12:11am UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503 "2021-12-23T00:11:05Z")\ **Posts on this page:** 17\ **Page:** 2
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [January 8, 2022, 9:14pm UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/21 "2022-01-08T21:14:37Z")
> [@OldGuy](#): > > > [@Dr.Strangelove](#): > > > > That’s just the thing. Even if you only consider the case where it stays at the apex, you have apparently broken time reversibility. The marble went up the dome–why doesn’t it come back down? > > As I said, I’m agreeing with Chronos. It doesn’t come back down, because it’s impossible for the marble to reach the apex in finite time. OK, but why? (I think that is part of the thought experiment). E.g., if h=\frac{2}{3}r^{3/2}, conservation of energy gives \ddot{r}=g\sqrt{r}. Let’s keep r positive and try to roll the ball from r(0)=1 up to r=0 at some finite time. I get something like r(t) = (1-\sqrt{\frac{g}{12}}t)^4 as a particular solution. But then it has zero velocity at the top. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [January 8, 2022, 9:15pm UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/22 "2022-01-08T21:15:43Z")
> [@OldGuy](#): > > It doesn’t come back down, because it’s impossible for the marble to reach the apex in finite time. The paper I linked to above has the equations. The differential equation for the radial acceleration is just: \frac{d^2r}{dt^2} = r^{1/2} One solution is just: r(t) = 0 But there’s an entire other class of solutions: r(t) = \frac{1}{144}(t-T)^4 \ for \ t \ge T “An object at rest” and so on are adequate heuristics most of the time, but really the machinery of Newtonian mechanics is in differential equations. And it’s clear that there are multiple valid solutions in this case. ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [January 9, 2022, 12:21am UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/23 "2022-01-09T00:21:54Z")
I agree those are mathematical solutions to the equation for some initial conditions; however, the initial conditions are are both r(t) = 0 and frac{dr(t)}{dt} = 0. So integrating once frac{dr(t)}{dt} = frac{3}{2} r^{3/2} + frac{dr(0)}{dt} and the second term is zero. It no longer integrates to the cited value of r(t). ETA: Sorry I guess I don’t know how to write the math. I copied what was in your post but it doesn’t work. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [January 9, 2022, 1:14am UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/24 "2022-01-09T01:14:51Z")
> [@OldGuy](#): > > It no longer integrates to the cited value of r(t). I think it does, or at least the problem is not with his integration [rather with lack of Lipschitz continuity of the force]: if r(0)=0 and v(0)=0 where v=dr/dt, then (at least where r\>0 to avoid any nasty singularity 🙂 🙂 ) dv/dt=\sqrt{r} yields v\,dv = \sqrt{r}\,dr so v^2 = \frac{4}{3}r^{3/2} + a constant, which we conveniently set to 0 because we want the velocity to vanish at r=0. Then we easily get solutions 144 r = {(t+C)}^4 after separating the variables a second time. Also, you can directly check that @Dr.Strangelove 's solution satisfies the equation d^2r/dt^2=\sqrt{r}, just like your solution r(t) = 0, and in both cases v(0)=0. It is clear that the surface is not smooth at 0 , though, which may largely be the “point” of this scenario. ---
**Author:** ![leahcim](https://avatars.discourse-cdn.com/v4/letter/l/b4bc9f/32.png) [@leahcim](https://boards.straightdope.com/u/leahcim)\ **Post date:** [January 9, 2022, 2:28am UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/25 "2022-01-09T02:28:39Z")
> [@Dr.Strangelove](#): > > “An object at rest” and so on are adequate heuristics most of the time, but really the machinery of Newtonian mechanics is in differential equations. And it’s clear that there are multiple valid solutions in this case. I looked at the paper, and have no doubt that for the given potential, the equations of motion do permit more than one solution passing through r = 0, r’=0. The potential was picked to not satisfy the conditions of the existence-uniqueness theorem for ODEs. I’d argue that the potential itself is unphysical, except as an approximation. The real dome is made up of real molecules so the forces are necessarily a sum of intermolecular forces which all obey the conditions of the E-U theorem, and since they’re all continuity/differentiability conditions, they also do so in aggregate. The lack of uniqueness comes from making an approximation to the actual potential that ignores the fine detail. ---
**Author:** ![k9bfriender](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/k9bfriender/32/3283_2.png) [@k9bfriender](https://boards.straightdope.com/u/k9bfriender)\ **Post date:** [January 9, 2022, 2:08pm UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/26 "2022-01-09T14:08:26Z")
Right, the cows are spheres. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [January 9, 2022, 2:13pm UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/27 "2022-01-09T14:13:16Z")
One I always loved was the Newtonian 5-body system, where one star gets gravitationally slung to infinity, in a finite time. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [January 9, 2022, 2:25pm UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/28 "2022-01-09T14:25:58Z")
Yeah. I read an article in Scientific American about that scenario, decades ago, and found it fascinating. > **[Zhihong Xia](https://en.wikipedia.org/wiki/Zhihong_Xia)** > > Pages for logged out editors learn more > > Zhihong "Jeff" Xia (Chinese: 夏志宏; pinyin: Xià Zhìhóng; born 20 September 1962 in Dongtai, Jiangsu, China) is a Chinese-American mathematician. > Xia received in 1982 from Nanjing University a bachelor's degree in astronomy and in 1988 a PhD in mathematics from Northwestern University with thesis advisor Donald G. Saari and thesis The Existence of the Non-Collision Singularities. From 1988 to 1990 Xia was an assistant professor at Harvard University... ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [January 9, 2022, 2:44pm UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/29 "2022-01-09T14:44:42Z")
And here’s the first page of the article, from 1991 [https://www.jstor.org/stable/24938845](https://www.jstor.org/stable/24938845) ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [January 9, 2022, 8:53pm UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/30 "2022-01-09T20:53:40Z")
I think that “marble on a dome” potential only “works” (i.e., takes a finite time for the marble to come to rest exactly on the peak) if the “marble” has zero size. Otherwise, you have a region on the top where it acts identically to a spherical marble on a conical point, which means that the actual shape of the potential function there is just spherical, and the time to come to rest is again infinite. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [January 9, 2022, 10:20pm UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/31 "2022-01-09T22:20:02Z")
> [@leahcim](#): > > The real dome is made up of real molecules so the forces are necessarily a sum of intermolecular forces which all obey the conditions of the E-U theorem, and since they’re all continuity/differentiability conditions, they also do so in aggregate. On one hand, that’s clearly true of the universe we live in. But on the other hand, it seems somehow unsatisfying to me. Instead of being ruled out by the basic laws, situations like this are only ruled out by the specific details of our universe. One has to go in manually and verify that the E-U theorem applies to all of the potentials that exist. Thought experiments usually don’t require that kind of unpleasant detail work. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [January 9, 2022, 10:46pm UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/32 "2022-01-09T22:46:53Z")
Related in some sense, there is, in (relativistic) physics a “cosmic censorship hypothesis”, the idea being that one would like to rule out a situation where there are singularities visible to observers. It is known at least that one can construct black holes with a Cauchy horizon inside where it is unpredictable what happens to a particle falling past it (cf [https://arxiv.org/pdf/1710.01722](https://arxiv.org/pdf/1710.01722)) ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [January 9, 2022, 11:45pm UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/33 "2022-01-09T23:45:45Z")
> [@Andy\_L](#): > > And here’s the first page of the article, from 1991 Are links to sci-hub allowed here? I can’t remember. In any case, if one _did_ have a means of accessing various academic archives, the DOI for this article is doi:10.1038/scientificamerican1291-144. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [January 9, 2022, 11:49pm UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/34 "2022-01-09T23:49:36Z")
Thanks. I was able to get the whole document from a computer with legitimate access to academic archives earlier today. ---
**Author:** ![Max\_S](https://avatars.discourse-cdn.com/v4/letter/m/46a35a/32.png) [@Max\_S](https://boards.straightdope.com/u/Max_S)\ **Post date:** [January 10, 2022, 7:16pm UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/35 "2022-01-10T19:16:29Z")
> [@Dr.Strangelove](#): > > The example I gave came from the paper [Causation as Folk Science](https://sites.pitt.edu/~jdnorton/papers/003004.pdf) by John D. Norton. Can someone explain to me how Prof. Norton derived the magnitude of the gravitational force tangential to the surface of the dome? He writes, > The dome of Figure 1a sits in a downward directed gravitational field, with acceleration due to gravity g. The dome has a radial coordinate r inscribed on its surface and is rotationally symmetric about the origin r=0, which is also the highest point of the dome. The shape of the dome is given by specifying h, how far the dome surface lies below this highest point, as a function of the radial coordinate in the surface, r. For simplicity of the mathematics, we shall set h = (2/3g) r^{\frac{3}{2}}. […] > > At any point, the magnitude of the gravitational force tangential to the surface is F = \frac{d(gh)}{dr} = r^{\frac{1}{2}} and is directed radially outward. There is no tangential force at r=0. Where did he get F = \frac{d(gh)}{dr}? Why is delta r, a radial coordinate, in the denominator? ~Max ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [January 10, 2022, 8:17pm UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/36 "2022-01-10T20:17:51Z")
Force is the derivative of potential energy with respect to distance (from the definition of work/energy as the integral of force over distance). The relevant distance is radial in this case. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [January 11, 2022, 12:19am UTC](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503/37 "2022-01-11T00:19:24Z")
P.S. gh is the standard expression for gravitational potential energy in a situation where gravity is constant . [Previous page](https://boards.straightdope.com/t/information-loss-in-theoretical-physics/956503.md?page=1)