Affine parameter Schwazschild

In summary, the coordinate 'r' serves as an affine parameter for radial null geodesics in the region exterior to the horizon for Schwarzschild spacetimes. This can be seen by considering the geodesic equation and calculating the Christoffel symbols, which result in the equation equating to zero. Additionally, the GRTensor package on Maple may give slightly incorrect Christoffel symbols for Schwarzschild spacetimes.
  • #1
LAHLH
409
1
Hi,

I've heard it said that for Schwarzschild spacetimes, then the coordinate 'r' is an affine parameter for radial null geodesics in the region exterior to the horizon. It seems weird to me that one of the coords is an affine parameter.

How can this be seen? I know that the radial null geodesics satisfy [tex] \tfrac{dt}{dr}=\pm \tfrac{1}{1-\tfrac{2M}{r} } [/tex], but what does this have to do with r being affine?
As far as I'm aware affine simply means related to the proper time linearly, i.e. [tex] \lambda=a\tau+b [/tex], and for any parameter related this way the geodesic equation (zero on the RHS) will be satisfied.

The only thing I could think of would be to take my [tex] \tfrac{dt}{dr}=\pm \tfrac{1}{1-\tfrac{2M}{r} } [/tex] and differentiate again, to get [tex] \tfrac{d^2t}{dr^2}=\mp\tfrac{2M}{(2M-r)^2 } [/tex]. Then look at the LHS of the geodesic equation [tex] \tfrac{d^2t}{dr^2}+\Gamma^{t}_{\mu\nu}\tfrac{dx^{\mu}}{dr}\tfrac{dx^{\nu}}{dr} [/tex] and calculate the Christoffel symbols and show this equation equates to zero.

For radial geo's this equation reduces to, [tex] \tfrac{d^2t}{dr^2}=\mp\tfrac{2M}{(2M-r)^2 } [/tex]. Then look at the LHS of the geodesic equation [tex] \tfrac{d^2t}{dr^2}+\Gamma^{t}_{tt}\left(\tfrac{dt}{dr}\right)^2+\Gamma^{t}_{rt}\left(\tfrac{dt}{dr}\right) +\Gamma^{t}_{tr}\left(\tfrac{dt}{dr}\right)+\Gamma^{t}_{rr}[/tex] which follows from dr/dr=1.

I find that [tex] \Gamma^{t}_{t r}=\tfrac{m}{((2m-r) r)} [/tex] and [tex] \Gamma^{t}_{r t}=-\tfrac{m}{((2m-r) r)} [/tex] and the other two zero. So LHS doesn't vanish this way, and there goes that idea...

My other thought is possibley to consider the magnitude of the tangent vector, for null geo's this should be zero?
 
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  • #2
In this case proper time is not a well defined concept, since you are considering a null curve. I cannot give you a formal definition, but I think of an affine parameter as something that parametrizes the curve under consideration in a good way - i.e. every point on the curve is in one-to-one correspondence with a single value of the parameter. Hope this helps..
 
  • #3
If you compute it carefully you'll get:
[tex] \frac{d^2 t}{d r^2} = \frac{\frac{2m}{r^2}}{\left( 1-\frac{2m}{r} \right)^2}[/tex]
[tex] \Gamma^{t}_{tt} =0[/tex],
[tex] \Gamma^{t}_{rt} = \Gamma^{t}_{tr} =
\frac{\frac{m}{r^2}}{\left( 1-\frac{2m}{r} \right)}[/tex],
[tex] \Gamma^{t}_{rr} =0[/tex]
so we have for [tex] \frac{d t}{d r} = - \frac{1}{1-\frac{2m}{r}} [/tex]:
[tex]
\frac{d^2 t}{d r^2} + \Gamma^{t}_{tt}\left(\tfrac{dt}{dr}\right)^2+\Gamma^{t}_{rt}\left(\tfrac{dt}{dr}\right) +\Gamma^{t}_{tr}\left(\tfrac{dt}{dr}\right)+\Gamma^{t}_{rr} = 0
[/tex]
 
  • #4
Oh, I must have dropped a sign (should have realized from torsion free condition anyway), so is this sufficient to say r is affine? I mean we also would have to show the r comp of geodesic is zero too I guess, but providing this is true is this all we need?

EDIT:

I used the package GRTensorII on Maple just to check these christoffel symbols, and it gave my original answers i.e. [tex] \Gamma^{t}_{tr}=-\Gamma^{t}_{rt} [/tex] instead of [tex] \Gamma^{t}_{tr}=+\Gamma^{t}_{rt} [/tex]. I find this strange now as Carroll shows the answer you stated, and obviously I now expect the symbol to be symmetric on lower indices not anti sym. strange
 
Last edited:
  • #5
Probably you have to show that it holds also for r component of geodesic (I don't
see the reason why proving this only for t component would be sufficient). It's
very simple and you can easily check it.
 
  • #6
I haven't had chance to check the r comp of geodesic yet, but just finding it odd that the GRTensor package on Maple seems to get such a simple christoffel symbol for schwarzschild slightly wrong.

Anyhow, so what I meant to say was, is it sufficient to show that the geodesic equation is satisfied with a zero LHS to prove something is an affine parameter, is this the way people would normally go about it?
 
  • #7
We call the curve [tex] t \rightarrow x^\mu(\tau) [/tex] a geodesic iff
[tex] \frac{d^2 x^\mu}{d \tau^2} +
\Gamma^\mu_{\alpha\beta}\frac{d x^\alpha}{d \tau}\frac{d x^\beta}{d \tau}
= \alpha \frac{d x^\mu}{d \tau}[/tex] for some function [tex] \alpha(\tau) [/tex].
If this equation if fulfilled with [tex]\alpha \equiv 0 [/tex] then we say that
geodesic is in affine parametrisation (or [tex]\tau [/tex] is a affine parameter
of this geodesic). So the answer to your question is yes.
 
  • #8
GRTensor uses a rather unusual notation for the Christofffel symbols. To get the "standard" christoffel symbols, use the following defintion:grdef(`CC{ ^a b c} := Chr{b c ^a}`);

and calculate the Christoffel symbols as CC, not Gamma.

See https://www.physicsforums.com/showpost.php?p=375565&postcount=6 (or the earlier posts in the same thread where I was once upon a time confused by the same issue).
 
  • #9
pervect said:
GRTensor uses a rather unusual notation for the Christofffel symbols. To get the "standard" christoffel symbols, use the following defintion:


grdef(`CC{ ^a b c} := Chr{b c ^a}`);

and calculate the Christoffel symbols as CC, not Gamma.

See https://www.physicsforums.com/showpost.php?p=375565&postcount=6 (or the earlier posts in the same thread where I was once upon a time confused by the same issue).


Thanks a lot for that, glad I found it here for Schw as I prob wouldn't have noticed for more complicated solns
 

1. What is the Affine parameter in the Schwarzschild metric?

The Affine parameter is a type of coordinate used in the Schwarzschild metric, which describes the geometry of spacetime around a non-rotating, uncharged black hole. It is a measure of the proper time experienced by an observer moving along a particular trajectory in this spacetime.

2. How is the Affine parameter related to the Schwarzschild radius?

The Affine parameter is directly proportional to the Schwarzschild radius, which is a measure of the event horizon of a black hole. This means that as the Affine parameter increases, an observer gets closer to the event horizon and experiences a stronger gravitational pull.

3. Can the Affine parameter be used to calculate the escape velocity of a black hole?

Yes, the Affine parameter can be used to calculate the escape velocity of a black hole. This can be done by setting the Affine parameter to infinity, which represents an infinite distance from the black hole, and using the equation for escape velocity in terms of the mass and radius of the black hole.

4. How is the Affine parameter related to the proper time of an observer?

The Affine parameter is a measure of the proper time experienced by an observer moving along a particular trajectory in the Schwarzschild metric. This means that as the Affine parameter increases, the proper time experienced by the observer also increases.

5. What is the significance of the Affine parameter in the Schwarzschild metric?

The Affine parameter is important in the Schwarzschild metric because it allows for the calculation of various physical quantities, such as proper time and escape velocity, in the spacetime around a black hole. It also allows for a more precise understanding of the effects of gravity on objects and observers near a black hole.

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