"Proper distance" in GR

yuiop

To start you off here are the equations for radial motion in Schwarzschild coordinates based on information from the mathpages website

http://www.mathpages.com/rr/s6-07/6-07.htm

1. Free fall from infinity with a given initial velocity V (including V=0)

The coordinate velocity of a free falling particle at r, with initial velocity of V at infinity is:

[tex]\frac{dr}{dt} = (1-2m/r) \sqrt{\frac{2m}{r}(1-V^2) +V^2} [/tex]

The time dilation factor for the falling particle is:

[tex]\frac{dt}{dtau} = \frac{1}{(1-2m/r) \sqrt{1 - V^2}} [/tex]

It follows that the proper velocity of a free falling particle in terms of the proper time of the particle is:

[tex]\frac{dr}{dtau} = \sqrt{\frac{2m}{r} + \frac{ V^2}{(1-V^2)}} [/tex]

2. Free fall from a given r value.

The coordinate velocity of a free falling particle at r that was initially at rest at R is:

[tex]\frac{dr}{dt} = \frac{(1-2m/r)}{ \sqrt{1-2m/R} } \sqrt{\frac{2m}{r} - \frac{2m}{R}} [/tex]

The time dilation factor for the particle dropped from R is:

[tex]\frac{dt}{dtau} = \frac{ \sqrt{1 - 2m/R}}{ (1-2m/r) } [/tex]

It follows that the proper velocity of a free falling particle dropped from R in terms of the proper time of the particle is:

[tex]\frac{dr}{dtau} = \sqrt{\frac{2m}{r} - \frac{2m}{R}} [/tex]

3. An linearly accelerating observer falling in the field (e.g. the magnitude of his acceleration is smaller than the inertial acceleration of the field).

This is more complicated. Some equations for this situation were given in #345 of this old thread: http://www.physicsforums.com/showthr...88#post2747788

[EDIT] The equations for a particle with initial velocity V have been edited to correct a major mistake in the calculations pointed out by Passionflower. [/EDIT]

PAllen

The definition of proper distance doesn't employ coordinate dependent quantities. However, this was in the section 'physical meaning'. In SR, one often talks about proper distance between two events being the distance you would measure in a frame in which the events are simultaneous. I was seeking to make an equivalent statement in GR, where the SR statement is totally inadequate (actually, it isn't adequate in SR if you allow accelerated frames, but you can trivially fix it by restricting to inertial) For GR and events on either side of e.g. a black hole, you need more of a fix. So my attempt at the best equivalent statement in GR is:

The proper distance between two events is equal to the 3-distance measured in a coordinate system containing a minimal geodesic in a hypersurface of simultaneity.

PAllen

The definition of proper distance doesn't use anything related to the NNT condition or any coordinates at all. Check the definition section again to see this. The NNT idea comes in with an attempt to say something limited about the extremal properties spacetime geodesics. I establish that without restrictions in nearby paths, no max/min statement is possible at all, even in SR. So I am only looking here to propose a restricted sense in which you can declare a max/min property.
I am not referring to these geodesics at all. Much later, in talking about physical meaning of proper distance I talk about 3-geodesics in a carefully chosen simultaneity slice. But in this section I am only talking about spacetime geodesics and spacetime paths.

This observation gets at a small omission in the extremal conjecture section, where an additional issue I dealt with later (in the "meaning of proper distance" section) I forgot to include in the geodesic extremal conjecture. I thought of this early today, but haven't had a chance to fix it. That, indeed, is related to what you say here.

I meant to specify that a spacelike geodesic is a local minimum related to NNT paths in a coordinate system that includes that geodesic as an NNT path. This is they way I did it later.

Then I think there is nothing wrong with my conjecture. I will see if I can still edit the original post, otherwise I will post a corrected version of the extremal conjecture.

PAllen

In reference to my long post #51, which I can no longer edit, I describe a limited statement about in what sense a spacelike geodesic can be considered minimal. The statement in that post is correct but worded in a coordinate dependent way. I realize a somewhat stronger, more coordinate independent statement is implicit, as follows (you need to read this in conjunction with post #51 for terminology):

Suppose there is a path p between two events with spacelike separation,
and interval(p) < interval(nearby geodesic between the same events). Then in every coordinate
system in which the geodesic is NNT, p will violate NNT.

pervect

The method you describe for distance is due to Born, it's the concept of using Born rigid motion to define length, though it's one I also like and have espoused in the past.

A more detailed calculation confirms to my satisfaction that the fermi-normal distance will give the same result as the measurement of distance via a Born rigid ruler.

(I felt I needed to confirm that the timelike worldlines of the accelerating observers at constant fermi-normal distance were orthogonal to the space-like geodesics along which the distance was measured. This will be true if the metric in these coordinates is diagonal away from the origin, which appears to be the case. Using the same series approach but including higher order terms, I convinced myself that metric was in fact diagonal.)

It is a proper distance in the sense that it's a distance measured along a specific curve. Passionflower's distance will also be a proper distance - if he ever gets around to specifying the curve along which it's measured, that is :-)

However, I don't believe that passionflower's distance will be equivalent to the distance measured by dropping a Born rigid ruler.

yuiop

Just a quick note to acknowledge that Passionflower pointed out a major mistake in my calculations for the motion of a falling body with an initial velocity at infinity in post #69. I have now edited that post and hopefully the equations are now correct. Thanks for checking Passionflower. It is good to know that someone is checking the posted equations so that they might be a useful and accurate reference to someone in the future .

yuiop

As far as I can tell, Pf's distance is the distance measured by a series of free falling observers each with their own clock and very short Born rigid ruler. The observers would have to synchronise clocks at infinity and then jump at regular intervals of say one second. At the time the leading observer arrives at the event horizon, each observer notes their spatial separation from their nearest neighbours at that time (according to their own local clock) and the distance to the event horizon according to a given observer at "that time" is the sum of all the individual measurements. The observers have to be very close together and the Born rigid rulers very short. Unlike Fermi normal observers and clocks, the Pf free fall observers are not physically connected in any way and do not experience proper acceleration. They will notice that their separation from their immediate neighbours is continually changing according to measurements using their Born rigid rulers or radar measurements. The Pf distance should also agree with the Schwarzschild coordinate distance and with what is measured by a free falling observer using a rolling wheel meter like the odometer of a car, as also pointed out by Pf. I am pretty sure the Pf measurement is closely related to the PG coordinate system. It would be interesting to check that out.

Passionflower

You are welcome yuiop, I definitely can use some practice in checking mistakes in formulas as the formulas I write myself are often full of mistakes.

I attempted to generalize the formulas to obtain the proper velocity of a free falling observer wrt a local stationary observer, in the attempt to generalize the, what Pervect calls, 'the passionflower distance'. Then perhaps we can do the same thing but then for the 'Fermi distance'.

Apparently the velocity function can be parametrized, if I am not mistaken such thing was developed by Hartle (or perhaps he copied it before, I don't know that).

Basically the proper velocity wrt a local stationary observer of a radially free falling observer can be described as:

[tex]
v=\sqrt {{\frac {{A}^{2}-g_{{{\it tt}}}}{{A}^{2}}}}
[/tex]

where:

[tex]
g_{{{\it tt}}}=1-2\,{\frac {m}{r}}
[/tex]

In the case the free fall is from infinity A simply becomes 1.
In the case the free fall has an initial velocity then A is calculated as:

[tex]
A={\frac {1}{\sqrt {1-{v}^{2}}}}
[/tex]

Now, if we assume the formula below as given by yuiop is both correct and not an approximation:

[tex]
\frac{dr}{dtau} = \sqrt{\frac{2m}{r} - \frac{2m}{R}}
[/tex]

then there is a catch.

For A smaller than 1 the free fall starts from a given height. I was trying to obtain the correct formula in order to plug in the right A in such a case and here is where I got some surprises (I assume for the greater minds on the forum this is yet another demonstration of my lack of understanding). I did not succeed in expressing this in terms of r only. Initially I was trying to reason that if in some way I could 'subtract' the escape velocity at the given R and convert that into A I would get the expected results. But that did not work, it turns out that A is no longer constant during the free fall from a given height, we could express it in the following way to get the desired results, but it is ugly:

[tex]
A = -{\frac {\sqrt { \left( -rR+2\,mR-2\,rm \right) R \left( -r+2\,m
\right) }}{-rR+2\,mR-2\,rm}}
[/tex]

This 'monstrosity' is not even real valued!

Now the interesting question is why, assuming again the formula based on this is exact, does the parameter depend on r? Perhaps I made a mistake or is perhaps the formula to obtain the proper velocity from a given R only an approximation?

For instance it is rather tempting to calculate A for a drop at h independently of r by using:

[tex]
A=\sqrt {1-2\,{\frac {m}{h}}}
[/tex]

The result is close to yuiop's formula but not identical. The 'shape' of the slabs look rather similar.

JesseM

Is it true that each hovering observer's local surface of simultaneity in their local inertial rest frame is parallel to a line of constant Schwarzschild time and varying radius? If so, maybe Passionflower's distance would be the proper distance along a single spacelike curve where the tangent at each point along the curve is parallel to the local surface of simultaneity of a freefalling observer at that point?

Passionflower

Right, I think I did that, but then the results do not seem identical to the formula you quoted (see the graph). Could you confirm? It looks like it is off by some linear factor. I probably made some mistake.

Hopefully we can work out the kinks because then we have a generic velocity formula for a free falling observer. Then it is simply a matter of finding the Lorentz factor based on the obtained velocity v(r) and divide (because it is length contraction! ) the integrand by the Lorentz factor to get the correct integrand to obtain a proper distance in all free falling radial scenarios (except for acceleration that is, but I think it is better to get the kinks out free falling motion first). By the way, the parametric approach has the advantage that we can extend things to a solution with a rotating black hole by applying some adjustments. As far as I am concerned that would then cover most areas where the notion of distance makes some sense, as the concept of proper (e.g. physical) distance in non-stationary spacetimes do not make much sense to me (except perhaps in things like FLRW models where distances are 'pumped up' in time). To me, but that is likely because I am simple minded, the generic distance discussion is rather academic for arbitrary non-stationary spacetimes as there is really not much to calculate.

By the way once this is sorted out it would be really interesting to lay out (in a separate topic) all the Doppler shifts (including a decomposition into kinematical and gravitational components) for all radial motion wrt a stationary local observer, an observer from infinity and of course wrt other free falling observers. As far as I know, no such generic formula is available.

yuiop

O.K. I have figured it now. The equation given by Hartle is not proper velocity (dr/ddtau), but the coordinate velocity (dr'/dt') according to a local stationary observer using his local stationary rulers and clocks. Proper velocity is the velocity using the proper time of the falling clock which is a different thing.

Starting with the Schwarzschild falling coordinate velocity I gave in #69:

[tex]
\frac{dr}{dt} = (1-2m/r) \sqrt{\frac{2m}{r}(1-V^2) +V^2}
[/tex]

and using dr'/dr = 1/sqrt(1-2m/r) and dt/dt' = 1/sqrt(1-2m/r) to relate local measurements of the stationary observer to the observer at infinity such that dr/dt = dr'/dt' (1-2m/r), the local coordinate velocity is:

[tex]
\frac{dr'}{dt'} = \sqrt{\frac{2m}{r}(1-V^2) +V^2}
[/tex]

This can be written using A = 1/sqrt(1-v^2) as:

[tex]
\frac{dr'}{dt'} = \sqrt{\frac{2m/r +V^2A^2}{A^2} }
[/tex]

[tex]\Rightarrow \frac{dr'}{dt'} = \sqrt{\frac{2m/r + A^2 - A^2 +V^2A^2}{A^2} } [/tex]

(because A^2-A^2 =0)

[tex]\Rightarrow \frac{dr'}{dt'} = \sqrt{\frac{2m/r + A^2 - A^2(1-V^2)}{A^2} } [/tex]

[tex]\Rightarrow \frac{dr'}{dt'} = \sqrt{\frac{2m/r + A^2 - 1 }{A^2} [/tex]

[tex]\Rightarrow \frac{dr'}{dt'} = \sqrt{\frac{ A^2 - (1 -2m/r) }{A^2} [/tex]

[tex]\Rightarrow \frac{dr'}{dt'} = \sqrt{\frac{ A^2 - g_{\it tt} }{A^2} [/tex]

Passionflower

Just to verify we are on the same page, you do not mean coordinate velocity in terms of Schw. coordinates right? Because clearly a local stationary observer would not measure such coordinates with local clocks and rulers. Also the free falling observer would measure the same velocity right, since this is simply the principle of relativity at work, right (e.g. time dilation, length contraction, locally Minkowskian)? But if so, then what is proper velocity in this context if you call this one coordinate velocity? Celerity?

Since:

[tex]
{dt \over d\tau} = {A \over g_{tt} }
[/tex]

It should be staightforward to convert this coordinate velocity to proper velocity.

By the way it would be helpful is someone could attach the relevant page from Hartle, unfortunately I do not have it: http://books.google.com/books?id=azZ...ed=0CC0Q6AEwAA

yuiop

I was just trying to make clear the distinction between proper velocity and coordinate velocity. Proper velocity (or celerity) is coordinate distance (which depends on the observer) divided by the proper time of the moving object. In post #69 I gave the proper velocity of the falling object according to the Schwarschild observer at infinity as:

[tex]
\frac{dr}{dtau} = \sqrt{\frac{2m}{r} + \frac{ V^2}{(1-V^2)}}
[/tex]

where tau is the proper time of the falling clock. This can be converted to the proper velocity as measured by a local observer at coordinate radius r by using dr'/dr = sqrt(1-2m/r) so that:


[tex]
\frac{dr'}{dtau} = \frac{1}{\sqrt{1-2m/r}} \sqrt{\frac{2m}{r} + \frac{ V^2}{(1-V^2)}}
[/tex]

and this can be simplified using A = 1/sqrt(1-V^2) and [itex]g_{tt} = (1-2m/r)[/itex] to:

[tex]
\frac{dr'}{dtau} = \sqrt{\frac{A^2 - g_{tt}}{g_{tt}}}
[/tex]

The equation for the local velocity of the falling object given by Hartle is obviously not the proper velocity.

Yes we can do it that way too. Starting with the Hartle equation for local velocity:

[tex]
\frac{dr'}{dt'} =\sqrt {{\frac {{A}^{2}-g_{{{\it tt}}}}{{A}^{2}}}}
[/tex]

the local proper velocity is:

[tex]\frac{dr'}{dtau} = \frac{dr'}{dt'}*\frac{dt'}{dt}*\frac{dt}{dtau} = \frac{A}{\sqrt{g_{tt}}} \sqrt {\frac {{A}^{2}-g_{tt}}{A^2}} = \sqrt{\frac{A^2 - g_{tt}}{g_{tt}}} [/tex]

Passionflower

Ok, it looks we are on the same page.

After some tinkering, the simplest generic formula I came up with is not the 'Hartle' approach but this:

[tex]
v_{(r)} = \sqrt {2m \left( {r}^{-1}-{{\it r_0}}^{-1}-{\frac { \left( -r+1
\right) {{\it v_0}}^{2}}{r}} \right) }
[/tex]

I worked out a position-velocity equivalent, and it turned out it is much simpler to express an initial velocity as a (virtual) position than vice versa.

Lorentz factoring the equation gives:

[tex]
{1 \over \gamma} =\sqrt {1-2\,m \left( {r}^{-1}-{r_{{0}}}^{-1}-{\frac { \left( -r+1
\right) {v_{{0}}}^{2}}{r}} \right) }
[/tex]

Multiplying 1/gamma by the integrand used by a stationary observer to obtain the distance for an arbitrary radial free falling observer gives:

[tex]
\int _{{\it r1}}^{{\it r2}}\!\sqrt {1-2\,m \left( {r}^{-1}-{{\it r_0}}^
{-1}-{\frac { \left( -r+1 \right) {{\it v_0}}^{2}}{r}} \right) }{\frac
{1}{\sqrt {1-2\,{\frac {m}{r}}}}}{dr}
[/tex]

A few plots:

yuiop

Before I check this out in detail, can you confirm that to use your formula, that if r0 is not infinite then v0 must be set to zero and if v0 is not zero then r0 has to be set to infinite?

Passionflower

Yes, as far as I understand both parts 'on' would lead to something unphysical.

yuiop

If I set v0 = 0 then the above equation reduces to:

[tex]
v_{(r)} = \sqrt {2m \left( {r}^{-1}-{{\it r_0}}^{-1}\right) }
[/tex]

which agrees with the mathpages equation I gave in #69 for the proper velocity dr/dtau:

Now if I set r0 = infinite in your equation, it reduces to:

[tex]
\frac{dr}{dt} = \sqrt{\frac{2m}{r}(1-v_0^2) +2mv_0^2}
[/tex]

which does not agree with either the mathpages coordinate velocity or proper velocity:

although it is similar to the coordinate velocity. More tinkering required I think

You did not specify if [itex]v_{(r)}[/itex] is intended to be a proper or coordinate velocity or if the observer that makes the measurement is local or at infinity.

[EDIT] Actually looking back I am guilty of the same thing in #69 and did not make it clear that all measurements in that post are according to the Schwarzschild observer at infinity.