Radial motion in the Schwarzshild spacetime

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1. May 23, 2014

Theactualbman

Hello,

The following has been confusing my friends and I, I want to make sure I have this clear as it is fairly elementary. (note set c = 1)

Ed is falling radially into a black hole, the Schwarzschild metric is:

ds2 = (1-2μ/r) dt2 - (1- 2μ/r)-1 dr2

his proper time is dτ2 = (1-2μ/r) dt2 - (1- 2μ/r)-1 dr2

If Simon is stationary and is coincident with Ed at some radius then he measures the proper distances and times given by:

drs = (1 - 2μ/r)-1/2 * dr and dts = (1 - 2μ/r)1/2 dt

I think that's all right. But then is Simon's metric dts2 - drs2 ≈ ds2 = (1-2μ/r) dt2 - (1- 2μ/r)-1 dr2 or are these really meant to be different bases?

Many thanks.

2. May 23, 2014

xox

correct (there are no rotational terms)

The above doesn't seem right. For Simon, $dr=0$.

3. May 23, 2014

Theactualbman

Yes that's what I thought, but if one solves the geodesic equations you find dr/dτ = -√2μ/r. If instead of perameterizing the metric in terms of τ, we instead use the proper time and proper distance of Simon then we find the corresponding derivative drs/dts = -√2μ/r, I found this in a bunch of textbooks (e.g. General Relativity (Hobson et al)). The real dilemma for me is whether the drs and dts correspond to different parameterizations of the metric, or if they're part of the same one I wrote above.

4. May 23, 2014

George Jones

Staff Emeritus
These relations apply to different situations.

Suppose that Simon is standing on a spherical shell that has r constant on its shell. and that two pennies plop down at Simon's. The two plops happen at the same place, dr = 0, but at different times. The second equation relates the time difference between plops according to Simon's watch to the difference in Schwarzschild coordinate time.

Now suppose that there is a second spherical shell just bellow Simon's spherical shell. Simon drills a hole in his shell, sticks a metre stick through the hole, and measures the spatial distance between the shells. The two positions of the shells against the metre stick are taken simultaneously, dt =0. The first equation relates the metre stick distance to the difference in Schwarzschild r.

5. May 23, 2014

xox

They do. You cannot combine them into one metric as you tried.

You cannot combine them into one metric as you tried.

6. May 23, 2014

Theactualbman

Okay, so then in Simon's small patch of the spacetime, he sees Ed go past at drs/dts.

How does Simon perameterize the invariant ds2. Is it even the same 'invariant' as Ed's?

And I still don't quite understand why you can't write

dts2 - drs2 ≈ ds2 = (1-2μ/r) dt2 - (1- 2μ/r)-1 dr2 , to my mind it follows by direct substitution and describes the flat tangent space to the point where Simon is.

(also, forgive me if I seem argumentative, I'm sure you're right, but I want this to be clear)

7. May 23, 2014

WannabeNewton

$d\tau^2 - d\rho^2$ is not Simon's metric (I am denoting proper time and proper distance by $\tau,\rho$). Why would you think this to be the case? $d\tau^2$ and $d\rho^2$ are obtained from two entirely different situations along Simon's world-line.

8. May 23, 2014

PAllen

If Simon set up local coordinates following the conventions used setting up standard inertial coordinates in SR, he would would end up with a local metric that would approach the Rindler metric very near him (actually, the Rindler metric translated to have an origin away from the horizon). If Ed did the same, his metric would very locally be the standard SR Minkowski metric. These statements are true independent of the global geometry (SC or otherwise). They are direct consequences of the definition of pseudo-Riemannian manifold, specifically that the tangent space is everywhere Minkowski space.

9. May 23, 2014

Staff: Mentor

Simon is not moving on a geodesic trajectory (though Ed is). He is not in free fall; he feels acceleration (he has to in order to stay at a constant altitude).

10. May 23, 2014

xox

That would mean that, according to your calculations, Simon measures the radial speed $v=\frac{1}{1-2 \mu/r} \frac{dr}{dt}$ for Ed. This is not only incorrect, it is also useless, since it is dependent on $\frac{dr}{dt}$.
The correct derivation is considerably more complicated, you can find it on page 27 of "Black Holes, an Introduction" by Raines and Thomas. I highly recommend this little book, it is excellent.

PS: the correct answer is :$v=\sqrt{\frac{\mu/r}{1-2 \mu/r}}$. I could derive it for you, if you are really interested but I think that you could benefit a lot more from buying the book.

11. May 23, 2014

Theactualbman

So Simon cannot construct a Minkowski metric around him since he isn't in freefall, that makes a lot more sense.

12. May 23, 2014

Staff: Mentor

He can, but it will only be valid in a small patch of spacetime around a particular event on his worldine. And his worldline will not look like a straight line in the Minkowski metric in that small patch; it will look like a hyperbola because he is accelerated (Ed's worldline will look like a straight line).

If you mean a metric in which Simon's worldline is a straight line, then yes, it can't be of the form you said. A metric in which Simon's worldline is a straight line will be, as PAllen pointed out, a Rindler metric.

13. May 23, 2014

George Jones

Staff Emeritus
This isn't correct. If Ed falls from rest from a large distance, then Ed's passing speed, as meausred by Simon, is
$$v= \sqrt{\frac{2 \mu}{r}}.$$

14. May 23, 2014

xox

The book I quoted seems to disagree. As far as I know, $\sqrt{\frac{2 \mu}{r}}$ is Ed's proper speed, not the coordinate speed as measured by Simon.

15. May 23, 2014

PAllen

If Simon sets up standard local coordinates, e.g. Fermi-Normal (which would take the form of a translated Rindler metric in this case), then the coordinate speed using these locally physically meaningful coordinates would match the proper speed. More importantly, this is the speed any reasonable experimental procedure would measure.

16. May 23, 2014

Theactualbman

$v=\frac{1}{1-2 \mu/r} \frac{dr}{dt}$ then gives that result doesn't it (applying the geodesic equations to Ed who follows a geodesic). Is it the case that in the neighbourhood of Simon the metric I suggest holds, but only extremely locally?

17. May 23, 2014

George Jones

Staff Emeritus
This means actualbman is considering what you (xox) call "proper speed". Note also that I wrote

"Ed's passing speed, as measured by Simon", whereas you did not qualify your speed. Unqualified, I took v to be physical speed. What else would would Simon measure? Coordinate speed? Why and how would Simon measure this?

I gave what Raine and Thomas call $v_{\mathrm{loc}}$, "the velocity of the object relative to the shell" on page 39 of their second edition. My first edition is at home, so I don't know the page in it.

18. May 23, 2014

Theactualbman

Also xox isn't the result you quoted for an observer at infinity?

19. May 23, 2014

xox

No, it isn't.

20. May 23, 2014

xox

I qualified it by referring the reader to page 27 , the 2006 edition.

I cannot locate anything like the above in my 2006 book.