Inconsistent forms of the metric in a uniform field

[bcrowell]

I would be interested in grappling with the geodesic equations of the PM and checking if they can offer us a coordinates by which the uniform acceleration is guaranteed everywhere!

The coordinates in which the metric is normally written are coordinates in which the proper acceleration of a particle released at rest is the same everywhere. See #79. Of course if you want that acceleration to *remain* constant forever (i.e., to be constant regardless of the particle's current velocity) then that's impossible.

But if I could catch the mainstream of the paper given at post #4 regarding the Petrov metric (PM) which is claimed to be the only rotating vacuum solution to Einstein field equations,

No, they don't claim that. There are many rotating vacuum solutions to the EFE. What is unique about the Petrov metric is its symmetry. It is "The only vacuum solution of Einstein’s equations admitting a simply-transitive four-dimensional maximal group of motions..."

 

[Altabeh]

I played around a little with the geodesic equations in the Petrov metric. Suppose you restrict yourself to a fixed z, and for convenience take an observer at an r such that \cos\sqrt{3}r=1, which means that the timelike coordinate is t. For a particle released at rest, the geodesic equations become
<br /> \ddot{t}=0<br />
<br /> \ddot{\phi}=0<br />
<br /> \ddot{r}=-\frac{1}{2}e^r \dot{t}^2<br />
where dots mean differentiation with respect to the affine parameter. If you take the affine parameter to be the particle's proper time, then \dot{t}=e^{-r/2} initially, and the particle accelerates toward lower values of r with a proper acceleration that is independent of r. If you go to a place where \cos\sqrt{3}r=-1, the timelike coordinate is again t, but I think the direction of time is reversed here as compared with at the place where \cos\sqrt{3}r=+1. Here the particle accelerates toward higher values of r. This seems to match up with what the Gibbons and Gielen paper describes about the geodesics: they oscillate back and forth in r.

So in this sense the Petrov metric seems to be a better realization of the uniform gravitational field than the Rindler coordinates, because the proper acceleration has the same magnitude everywhere. On the other hand, it does have strange properties like CTCs, and that's undesirable.

[EDIT] I think the constancy of the proper acceleration follows from the fact that in addition to the three obvious Killing fields, there is a fourth one involving r. If I've got that right, then the restriction of my calculation above to certain values of r can be relaxed, and the result still holds.

If your calculations are all correct, then the assumption

\dot{t}=e^{-r/2}

is only in accordance with the first geodesic equation if one of the two conditions below holds:

1) r is taken to be a constant, so the motion is much restricted,
2) the proper velocity of particles along r is zero which is the case when an instantaneous comoving observer measures stuff regarding the motion of particles.

Of course the first condition would not be regarded as a tool of providing a constant proper acceleration globally. So talking about the other condition, and yet highly unphysical one, I don't see any better situation than when we were dealing with the Rindler metric because this is again dependent on position which is cured by setting the initial time of the observer's clock at the beginnig of the motion. And about the motion, there is no such thing along r though for an instantaneously at rest comoving observer along r at a constant z but with a varying \phi, one can have the proper acceleration \ddot{r} observed instantaneously and uniformly! But remember that this approach, I think, is so insubstantial in the context of global constancy of proper acceleration: we are actually binding time up into space through \dot{t}=e^{-r/2} in a way to have a uniform acceleration observed by some observer being instantaneously at rest while this is not essentially happening in the spacetime. I mean if you have z also involved as a freely varying coordinate, the situation gets much complicated and requires much more things that we need to mean in essence by 'global discussion' of the whole scenario than now!

Keeping in mind the global discussion, how one would show me all four elements can be adjusted somehow so as to make thee proper acceleration uniform in Petrov metric?

AB

 

[bcrowell]

If your calculations are all correct, then the assumption

\dot{t}=e^{-r/2}

is only in accordance with the first geodesic equation if one of the two conditions below holds:

1) r is taken to be a constant, so the motion is much restricted,
2) the proper velocity of particles along r is zero which is the case when an instantaneous comoving observer measures stuff regarding the motion of particles.

As stated explicitly in the text that you quoted, the result only holds for a particle that is instantaneously at rest. Please see my #37, in which I pointed out to you that I had already explicitly stated this twice, in #19 and #26.

Keeping in mind the global discussion, how one would show me all four elements can be adjusted somehow so as to make thee proper acceleration uniform in Petrov metric?

No adjustment is necessary. This was the result of the calculation in #79.

Folks, this has been a very enjoyable thread, but I think I'm going to stop following it now. Thanks, everyone, for sharing your valuable and insightful comments!

 

[Altabeh]

Let me give you a vivid picture of the globally uniform gravitational field and how it can be realized:

1- The metric must be Ricci-flat or a vacuum solution to the Einstein field equations;
2- In Cartesian coordiantes (t,x,y,z), the geodesic equations have the form

\ddot{t}, \ddot{x}, \ddot{y} and \ddot{z} must all be constant.
Or the Christoffel symbols be all constant.

It is not impossible to look for such a metric and I have some plan to do find one! I'll give it a go someday I find a little free time!

AB

 

[Altabeh]

As stated explicitly in the text that you quoted, the result only holds for a particle that is instantaneously at rest. Please see my #37, in which I pointed out to you that I had already explicitly stated this twice, in #19 and #26.

My bad, I didn't see 'em!

No adjustment is necessary. This was the result of the calculation in #79.

If not take instantaneously at rest as an "adjustment"!

Folks, this has been a very enjoyable thread, but I think I'm going to stop following it now. Thanks, everyone, for sharing your valuable and insightful comments!

Why? Anyways it's been too enjoyable for me, too and hope you are not leaving us forever!

AB

 

[Mentz114]

I'm signing off too with thanks to all posters. I have learned some things I should have known. I conjecture that -

1. The Rindler coords have nothing to do with gravity.
2. They are not global but apply locally to an accelerating observer in Minkowski space-time
3. The parameter 'a' is related to the observer's proper acceleration.
4. So far no-one has come up with a uniform gravitational field, either in vacuum or matter.