# Tidal Effects of Non-Uniform Gravitational Fields in Free-falling Coaches

1. Nov 29, 2007

### dx

theres a problem in wheeler's spacetime physics (exercise 2-9) which goes like this : you are in a horizontal railway coach launched upwards from the surface of the earth. there are two ball bearing at opposite ends of the coach. can you distinguish whether the coach is rising or falling by observing the ball bearings from inside the coach?

so this is what i thought : there will be tidal acceleration between the two ball bearings the whole time during the free fall since the gravitational field is not perfectly uniform. the further away you are from the earth, the smaller the tidal accelerations will be. so as you are rising, you will be able to observe that the tidal accelerations decrese, reach a minimum at the top, and increase again as you fall back to the earth. so we must be able to tell whether we are rising or falling.

the answer at the back of the book says that you cant distinguish between rising and falling by observing the ball bearings.

where did i go wrong?

2. Nov 29, 2007

### Chris Hillman

A Sophisticated Reponse I

You should cite this as Taylor and Wheeler, Spacetime Physics, Exercise 2-9. This classic textbook has a coauthor, and the problem in question might well have been written by Edwin Taylor, not John Wheeler!

[EDIT: dx should probably ignore the following, but PF readers who have more familiarity with elementary gtr might benefit from this post. Sorry, dx I should have said that in the original version of this post!]

The tidal tensor (in either Newtonian gravitation or gtr) in this situation can be put in the form (components taken wrt an appropriate coframe field or orthonormal basis of one-forms) $m/r^3 \, \operatorname{diag} (-2,1,1)$, or tidal tension along a line oriented toward the massive object and tidal compression orthogonal to this line. As I have often pointed out, if you are in a free-falling spaceship in the vacuum field produced by an isolated massive object, this means that in principle you can, without looking outside the window of your spaceship, align a rod so that one end points at the source of the field, but you can't tell which end! Using these remarks you can probably set up a detailed thought experiment and compare orders of magnitude.

Last edited: Nov 30, 2007
3. Nov 29, 2007

### dx

sorry but i dont really know much about gtr, im a beginning undergraduate student. could you tell me where my argument went wrong.

4. Nov 29, 2007

### pervect

Staff Emeritus
I think your argument that you can use the tidal force to determine the distance is correct. A single observation of the tidal force will not give you any information on whether you are rising or falling, but a sequence of observations will give you information on distance vs time.

I'm not quite sure what Wheeler had in mind with this question. It might have something to do with the interesting result that in the Schwarzschild geometry, the radial component of your velocity has no effect whatsoever on tidal forces (i.e. any of the components of the Riemann in your frame field).

Last edited: Nov 29, 2007
5. Nov 30, 2007

### Chris Hillman

A Sophisticated Response II

[EDIT: I should have started by saying: dx should probably ignore the following, but PF readers who have more familiarity with elementary gtr might benefit from this post--- sorry, dx, if I had been paying attention I would have noticed how unreasonable it was for me to write as if you could be expected to follow what I said here! So don't worry if you don't follow a word of it.]

My first response was probably too sketchy even for those PF readers who have taken more gtr than dx has, so let me try to elaborate a bit for the possible benefit of such readers.

In gtr the tidal tensor appears as one of the tensors in the Bel decomposition of the Riemann tensor wrt a timelike congruence (possibly having nonzero vorticity) and it can be written
$$E [\vec{X} ]_{ab} = R_{ambn} \, X^m \, X^n$$
Then the Jacobi geodesic formula (see for example Eq. (8.43) in MTW) becomes:
$$\frac{D^2 \eta^a}{ds^2} = - {{E[\vec{X}]}^a}_b \, \eta^b$$
Here, $\vec{\eta}$ is a spacelike seperation vector between two nearby timelike geodesics, and where $\vec{X}$ is the tangent vector to the world line of our spaceship (railway car).

Assuming the form of the tidal tensor which I gave in my Post #2 (and which takes the same form in either gtr---- using a Schwarzschild vacuum model--- or in Newtonian theory; deriving the latter claim is easy, in fact I have taught it to first year calculus students), let us consider a thought experiment conducted over a short time period inside our spaceship when it is at distance $r=r_0$ from an isolated massive source of mass m, in which a cloud of test particles dropped from rest at time zero. We have the simple initial value problem (in either Newtonian gravitation or gtr):
$$\begin{array}{ccc} \ddot{x} = 2 \, m/r_0 \, x, & \ddot{y} = -m/r_0 \, y, & \ddot{z} = -m/r_0 \, z, \\ x(0) = x_0, & y(0) = y_0, & z(0) = z_0, \\ \dot{x}(0) = 0, & \dot{y}(0) = 0 & \dot{z}(0) = 0 \end{array}$$
where we regard $r_0$ as well as m as a constant over the duration of the experiment. This has the solution
$$\begin{array}{rcl} x(s) & = & x_0 \, \cosh \left( \sqrt{2m/r_0} \, s \right) \\ y(s) & = & y_0 \, \cos \left( \sqrt{m/r_0} \, s \right) \\ z(s) & = & z_0 \, \cos \left( \sqrt{m/r_0} \, s \right) \end{array}$$
which to second order in the elapsed time is
$$\begin{array}{rcl} x(s) & = & x_0 \, \left(1 + \frac{2 \, m}{r_0} \cdot \frac{s^2}{2} \right) \\ y(s) & = & y_0 \, \left(1- \frac{m}{r_0} \cdot \frac{s^2}{2} \right) \\ z(s) & = & z_0 \, \left(1-\frac{m}{r_0} \cdot \frac{s^2}{2} \right) \end{array}$$
Thus, the particles diverge in the x direction and converge orthogonally; a thin spherical shell of test particles is deformed into a prolate ellipsoidal shell with the long axis oriented parallel to the x direction. The fact that the tidal tensor is traceless (which in both Newtonian gravitation and gtr is equivalent to choosing a vacuum solution) ensures that the volume inside such a shell does not change over a short duration experiment, although its shape is deformed.

Notice that, in our short duration experiment, the behavior of the test particles is symmetrical wrt +/- x direction. Furthermore, our model is symmetrical under time reversal. Notice also that I wrote the second order approximation to show that there is a huge conceptual difference between the Newtonian view (according to which tidal forces are acting as given by the tidal tensor) and the gtr view (according to which the world lines of the test particles are timelike geodesics and thus are subjected to no physical force; in gtr all gravitational phenomena are modeled using curvature, although whenever gtr and Newtonian theory give the same results, to some approximation, one can interpret geodesic deviation in quasi-Newtonian fashion).

(Pedantic caveat: to save the trouble of worrying, we can assume that the test particle exhibit nonrelativistic motion during the duration of our thought experiment, so we don't need to worry about "proper time" versus "time".)

The problem as dx stated it is perhaps not perfectly clear, but as I interpret it, the idea is to get the student to analyze this experiment and to draw these conclusions:
• without looking out the window of the spaceship, using one of our short duration experiments you can orient a rod inside the spaceship such that one of its ends points at the massive object--- but you can't say which end, at this order of approximation!,
• the behavior of the cloud of test particles is symmetric wrt the time of dropping the cloud of test particles, at this order of approximation,
• if over a much longer period of time than the brief duration of our thought experiment, the spaceship rises radially and then falls back, and if we repeatedly perform short duration experiments of the kind modeled above, then there is no way to distinguish between the event "$r=r0$ and increasing" versus the event "$r=r0$ and decreasing", at this order of approximation, except by examining the changing tidal forces inferred over a long term period from numerous short duration experiments as per above.
IOW, if you imagine repeatedly performing short duration experiments as above, you can indeed in principle track the evolution over time of the tidal tensor, and you can infer in which direction the source must lie, but you can't tell "up" from "down" or "going up" from "going down" from a single such experiment.

Last edited: Nov 30, 2007
6. Nov 30, 2007

### dx

so youre saying the answer at the back is wrong?

7. Nov 30, 2007

### pervect

Staff Emeritus
I'm thinking that Wheeler probably meant to say that you couldn't determine your velocity from a single measurement at one point in time of the tidal force and that you've interpreted his question differently than he intended.

8. Nov 30, 2007

### dx

This is what he says : "can you distinguish whether the coach is rising or falling with respect to the surface of Earth solely by observing the ball bearings from inside the coach?"

9. Nov 30, 2007

### Chris Hillman

An ambiguous problem with two distinct "correct answers"

Hi again, dx, it should be "safe" to read this post and I hope it will help.

By "he", I trust you mean "one or both of the two authors of the book coauthored by Edwin F. Taylor & John A. Wheeler, Spacetime Physics".

I see you are taking me at my word (re my suggestion that you ignore my over-sophisticated responses), which is probably a good thing, but at the end of my Post #5 I did suggest what I think is the solution to this mystery:

I guess that the intended interpretation was that when the coach is at some distance $r_0$ from the Earth, you perform a short experiment in which you allow ball bearings to roll freely at some time and try to deduce whether the coach is "going "up" or "going down". (Over the duration of this short experiment, the distance to the Earth is approximately constant.) The answer is that if only perform one such brief experiment, you can't tell the difference from "going up" and "going down", but if you know the mass of the Earth m and if you assume tidal forces of form $m/r_0^3 \, \operatorname{diag}(-2,1,1)$, where the distance to the Earth $r_0$ is approximately constant over the time of the brief ball-bearing experiment, then you in principle you can deduce $r_0$ from the motion of the ball bearings. (You can also deduce that "one end" of a certain line is pointing in the direction of the Earth, but you can't say which end!)

However, if you perform many such experiments and track how the tidal forces inside the coach evolve over time--- this is the interpretation you gave to the problem--- then we all agree that you can in principle deduce whether you are "going up" or "going down" because the tidal forces are decreasing or increasing respectively so you can deduce that the distance to the Earth is increasing or decreasing respectively. (And you can deduce the direction of the Earth.)

IOW, if one interprets the problem as I believe Taylor & Wheeler intended, the answer in the back is correct. If one interprets the problem the way you did (which I think is understandable from the statement which you quoted!), then we all agree that your answer is correct.

Or in short, ditto pervect

Last edited: Nov 30, 2007
10. Nov 30, 2007

### dx

ah ok thanks :)

11. Nov 30, 2007

### Chris Hillman

Incidently, re your avatar (from the book by Peitgen?), have you heard of John Hubbard (Math, Cornell) of "Hubbard and Douady" fame?

[Edit: for a full citation to "the book by Peitgen", see my Post #14 below.]

Last edited: Nov 30, 2007
12. Nov 30, 2007

### dx

please correct me if im wrong. this is what i understood from what you said : saying that the during the short experiment the distance from the earth is approximetely constant is like assuming that as far as your experiment is concerned, the region in spacetime in which you perform it is approximately flat. is it like assuming that your frame is inertial in that small region of spacetime?

13. Nov 30, 2007

### dx

oh, no i got the avatar off google. ya i have a book by a john hubbard called vector calculus, linear algebra and differential forms. dont know if hes the same john hubbard that youre talking about.

14. Nov 30, 2007

### Chris Hillman

Two time scales, two questions having distinct correct answers

dx asked whether

The spacetime curvature measurably affects the motion of the ball bearings during this short experiment, so no. Here "short duration" just means "sufficiently short that we can neglect the fact that r is either increasing or decreasing with time".

The world line of the (centroid of the) spaceship itself is treated as a timelike geodesic, but the world lines of the ball bearings are also modeled as timelike geodesics.

Maybe this will help: on the time scale of a day, the Earth rotates about its axis once. On the time scale of a year, the Earth traverses its (approximately periodic) orbit once. On the time scale of a day, we can appproximate the motion of the stars and most planets by simply spinning the "celestial sphere" wrt an observer at the center. On the time scale of a year, we have a much more complicated scenario in which planets may appear to momentarily stop and then "reverse their motion". Similarly, there are long and short time scales in the railway coach scenario. I believe that the authors of your textbook intended that students interpret the problem in terms of the short timescale, but it was imprecisely worded and you interpreted it in terms of the long timescale. These two time scales lead to distinct questions with distinct answers. That is why your answer (to a question about the long time scale behavior), and the books answer (to a question about the short time scale behavior) are both correct.

I hopefully inquire: have you encountered the Lemaitre coordinate chart for the Schwarzschild vacuum? If so, I can show how to investigate the effect of trying to allow for the fact that the distance is in fact changing over time, to see how the ball bearings behave in a more sensitive experiment conducted inside our spaceship (coach).

Sure, but I think it is originally from the book by H.-O. Peitgen and P.H. Richter, The beauty of fractals: images of complex dynamical systems, Springer, 1986.

The very same. Not only that, but I was going to recommend another book coauthored by him: John H. Hubbard, Beverly H. West, Differential equations: a dynamical systems approach, Springer, 1991.

[EDIT: Yup, add me to those who have fallen victim to the "misreading virus". How [thread=200063]ironic[/thread] is that? Sorry, dx, I wrongly accused you!]

Anyway, the point is that differential forms are a student's best friend

Last edited: Nov 30, 2007
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