Affine parametrization for null geodesic?

[stevendaryl]

The geodesic equation for a path $X^\mu(s)$ is:

$\frac{d}{d s} U^\mu + \Gamma^\mu_{\nu \tau} U^\nu U^\tau = 0$

where $U^\mu = \frac{d}{ds} X^\mu$

But this equation is only valid for affine parametrizations of the path. For a timelike path, being affine means that the parameter $s$ must be linearly related to the proper time $\tau$:

$s = A + B \tau$

But what is the constraint on the parameter $s$ when the path $X^\mu(s)$ is a null path (that is, $g_{\mu \nu}U^\mu U^\nu = 0$)?

 

[TrickyDicky]

The geodesic equation for a path $X^\mu(s)$ is:

$\frac{d}{d s} U^\mu + \Gamma^\mu_{\nu \tau} U^\nu U^\tau = 0$

where $U^\mu = \frac{d}{ds} X^\mu$

But this equation is only valid for affine parametrizations of the path. For a timelike path, being affine means that the parameter $s$ must be linearly related to the proper time $\tau$:

$s = A + B \tau$

But what is the constraint on the parameter $s$ when the path $X^\mu(s)$ is a null path (that is, $g_{\mu \nu}U^\mu U^\nu = 0$)?

None, other than the geodesic equation for null paths itself.

 

[Bill_K]

The definition of a null geodesic is that there exists a parameter s such that the geodesic equation holds. (Once you have one, it's easy to show that s' = as + b is another.)

 

[stevendaryl]

The definition of a null geodesic is that there exists a parameter s such that the geodesic equation holds. (Once you have one, it's easy to show that s' = as + b is another.)

Hmm. But what is the significance of the parameter s, since it's not proper time? I guess in flat spacetime it could very well be coordinate time.

 

[WannabeNewton]

It has no significance, it's just a parameter. It can't really have any significance because of the gauge freedom mentioned by Bill.

 

[stevendaryl]

It has no significance, it's just a parameter. It can't really have any significance because of the gauge freedom mentioned by Bill.

Why do you think that the freedom $s' = a + b s$ means that $s$ has no meaning? Such a change only amounts to a choice of a scale and a choice of a zero. In my mind, such changes don't have much significance.

 

[TrickyDicky]

Hmm. But what is the significance of the parameter s, since it's not proper time? I guess in flat spacetime it could very well be coordinate time.

The choice is infinite. What significance would you have in mind?
And of course it could be coordinate time but why only in Minkowski spacetime?

 

[stevendaryl]

The choice is infinite. What significance would you have in mind?
And of course it could be coordinate time but why only in Minkowski spacetime?

Well, in the case of slower-than-light geodesics, the affine parameter has a clear physical meaning: The change in the affine parameter from point A to point B is proportional to the elapsed time on an ideal clock traveling from A to B along that geodesic.

 

[stevendaryl]

The choice is infinite. What significance would you have in mind?
And of course it could be coordinate time but why only in Minkowski spacetime?

In curved spacetime with arbitrary coordinates, coordinate time is not necessarily an affine parameter.

 

[TrickyDicky]

In curved spacetime with arbitrary coordinates, coordinate time is not necessarily an affine parameter.

Any parameter that obeys the geodesic equation(with the affine connection) is by definition an affine parameter.

 

[stevendaryl]

Any parameter that obeys the geodesic equation(with the affine connection) is by definition an affine parameter.

Yes, you've already said that. The question is: what is the meaning of the affine parameters?

Here's another way to frame the question: You have a geodesic that passes through events A, B, and C. If it is the case that the change in the affine parameter in going from A to B is the same as the change in the affine parameter in going from B to C, then that fact is independent of the choice of the affine parameter. So that fact is a coordinate-independent and parametrization-independent fact about the geodesic. In the case of timelike geodesics, it has a clear meaning: it means that the proper time in going from A to B is the same as the proper time in going from B to C. In the case of lightlike geodesics, what does it mean? The claim that it doesn't mean anything seems wrong. At least the reasoning that it can't mean anything because it's dependent on an arbitrary parametrization is wrong, since this fact is independent of parametrization.

 

[PAllen]

Yes, you've already said that. The question is: what is the meaning of the affine parameters?

Here's another way to frame the question: You have a geodesic that passes through events A, B, and C. If it is the case that the change in the affine parameter in going from A to B is the same as the change in the affine parameter in going from B to C, then that fact is independent of the choice of the affine parameter. So that fact is a coordinate-independent and parametrization-independent fact about the geodesic. In the case of timelike geodesics, it has a clear meaning: it means that the proper time in going from A to B is the same as the proper time in going from B to C. In the case of lightlike geodesics, what does it mean? The claim that it doesn't mean anything seems wrong. At least the reasoning that it can't mean anything because it's dependent on an arbitrary parametrization is wrong, since this fact is independent of parametrization.

Adding to this, for a spacelike geodesic, it means the proper distance is the same (given A, B, C close enough that the goedesic is unique). I think this is a really good question. I don't know a good answer.

 

[Bill_K]

Adding to this, for a spacelike geodesic, it means the proper distance is the same (given A, B, C close enough that the goedesic is unique). I think this is a really good question. I don't know a good answer.

I think this is the answer, although maybe not a good one: it's the rescaled limit of the proper distance along neighboring non-null geodesics.

 

[TrickyDicky]

Or to relate it to proper time, it's the rescaled limit by a factor that approaches asymptotically infinite of the proper time as it tends to zero, but this was discussed and this interpretation in the form of limits was given in a thread in which stevendaryl, PAllen and I were participating, about a year or two ago. My impression was this was settled then for stevendaryl.
I think it's more physical and more related to GR to use the timelike geodesics approximation rather than the spacelike ones.

 

[stevendaryl]

Or to relate it to proper time, it's the rescaled limit by a factor that approaches asymptotically infinite of the proper time as it tends to zero, but this was discussed and this interpretation in the form of limits was given in a thread in which stevendaryl, PAllen and I were participating, about a year or two ago. My impression was this was settled then for stevendaryl.
I think it's more physical and more related to GR to use the timelike geodesics approximation rather than the spacelike ones.

I had completely forgotten the previous thread. My apologies for the duplication.

 

[yuiop]

In the Schwarzschild metric the coordinate radial velocity of a photon is $dr/dt = \pm(1-2m/r)$. If we take the indefinite integral wrt to r of the inverse velocity (dt/dr), we obtain the null geodesic equation:

$t = \pm (2m \ln(r-2m) + r )+ C$

where C is the constant of integration. How do we relate this to $a +b\tau$ ? In the case of a lightlike path, $\tau=0$ and presumably a is the constant of integration?

If both the positive and negative solutions are plotted, then curves represent a 2 dimensional future and past light cone for the event at the intersection. This intersection event conveniently provides a way to uniquely identify the light cone in terms of the radius and time of the intersection event. This is similar to the way that the timelike path of a free falling particle can be readily identified by the radius and time coordinates of its apogee. An exception is when a free falling massive particle has a non zero velocity at infinity and then the intersection of the ingoing and outgoing paths is used.

 

[TrickyDicky]

I had completely forgotten the previous thread. My apologies for the duplication.

No need to apologise, it was quite a bit ago and it is a valid question, some questions are recurrent which mean their resolution is probably not completely satisfactory.
All this reminds me of a recurrent theme in this subforum, people often come here asking about a photon's point of view, with the intuition that a photon "sees time still" and distances vanishing. Of course they are adviced to abandon those views as it is "meaningless" and in that sense sort of "forbidden" to adopt the frame of a photon in physics, which is true in a certain stric way of looking at it.
Still if one uses the limit of timelike geodesics concept to give physical content to the affine parameter s of null paths one is taking that "meaningless" photon's view only in a more sophisticated claculus-way using limits. So I would say this way of giving a physical significance to the affine parameter of null paths is far from being acceptable in PF.

 

[TrickyDicky]

In the Schwarzschild metric the coordinate radial velocity of a photon is dr/dt = (1-2m/r). If we take the indefinite integral wrt to r of the inverse velocity (dt/dr), we obtain the null geodesic equation:

$t = 2m \ln(r-2m) + r + C$

where C is the constant of integration. How do we relate this to $a +b\tau$ ? In the case of a lightlike path, $\tau=0$ and presumably a is the constant of integration?

As previously commented we can't use $\tau=0$ in the null case as parameter, you are using t and you can arbitrarily scale t and add a constant to it and it is still a valid affine parameter.

 

[Bill_K]

In the Schwarzschild metric the coordinate radial velocity of a photon is dr/dt = (1-2m/r). If we take the indefinite integral wrt to r of the inverse velocity (dt/dr), we obtain the null geodesic equation:

$t = 2m \ln(r-2m) + r + C$

where C is the constant of integration. How do we relate this to $a +b\tau$ ? In the case of a lightlike path, $\tau=0$ and presumably a is the constant of integration?

The energy integral is E = (1 - 2m/r)(dt/ds). [Making it obvious that t is NOT an affine parameter!]
Combine this with dr = (1 - 2m/r) dt and you get E = dr/ds, or s = r/E.

Strange-looking, but over the extent of a radial null geodesic, r may be taken as the independent variable, and is an affine parameter.

 

[TrickyDicky]

The energy integral is E = (1 - 2m/r)(dt/ds). [Making it obvious that t is NOT an affine parameter!]
Combine this with dr = (1 - 2m/r) dt and you get E = dr/ds, or s = r/E.

Strange-looking, but over the extent of a radial null geodesic, r may be taken as the independent variable, and is an affine parameter.

Well in the Schwarzschild geodesic equation for null paths can't you make E=dr/(1-2m/r)ds?

 

[Bill_K]

Well in the Schwarzschild geodesic equation for null paths can't you make E=dr/(1-2m/r)ds?

No, of course not. The energy integral is a direct consequence of the geodesic equations. I can't replace it with something else, E would no longer be a constant or the path would no longer be a geodesic.

 

[TrickyDicky]

No, of course not. The energy integral is a direct consequence of the geodesic equations. I can't replace it with something else, E would no longer be a constant or the path would no longer be a geodesic.

No, it would be the same E, I meant that in the equation for null geodesics using the Schwarzschild metric the dt2 term is made equal to the dr2-dΩ2 term. t is an independent variable in static spacetime.

 

[Bill_K]

t is an independent variable in static spacetime.

There's nothing that forces you to this choice. The independent variable is whatever you choose it to be. But the affine parameter is r in any case, or a linear function of r.

You could put everything in terms of t and use that as the independent variable, but the solution of the geodesic equation is t(r) = 2m ln (r - 2m) + r, and to write r(t) you'd have to invert this equation, which can't be done in closed form. So it's easier to leave things in terms of r.

Also note that t has a problem at the horizon r = 2m, while r does not. The ingoing null geodesics go all the way to r = 0, and r is valid as an affine parameter all the way. Same with the outgoing null geodesics if you want to use the minus sign and do them. [Actually, I guess + is outgoing, - is ingoing.]

 

[TrickyDicky]

There's nothing that forces you to this choice. The independent variable is whatever you choose it to be. But the affine parameter is r in any case, or a linear function of r.

You could put everything in terms of t and use that as the independent variable, but the solution of the geodesic equation is t(r) = 2m ln (r - 2m) + r, and to write r(t) you'd have to invert this equation, which can't be done in closed form. So it's easier to leave things in terms of r.

Of course nothing forces you to do it, the debate was whether using the coordinate time as afine parameter can be done at all in null geodesics, and the fact is that in static spacetimes it can be done as Padmanabhan shows in his GR text:
"In the case of null geodesics in a static spacetime one can introduce another variational principle, which is a generalization of Fermat’s principle to curved spacetime. Consider all null curves connecting two events P and Q in a static spacetime. Each null curve can be described by the three functions xα(t) and will take a particular amount of coordinate time Δt to go from P to Q. We will now show that the null geodesic connecting these two events extremizes Δt. To do this, we shall change the independent variable in Eq. (4.67) from the affine parameter λ to the coordinate time t by using the relation 0=dt2 +gαβ/g00dxα dxβ. (4.75) ...The Fermat principle is now equivalent to the statement that such a gravitational field acts like a medium with a refractive index n(x)=f(x)/ |g00(x)|. In addition to the bending of light, such an effective refractive index will also lead to a time delay in the propagation of light rays. This delay, called Shapiro time-delay has been observationally verified."

 

[PAllen]

Of course nothing forces you to do it, the debate was whether using the coordinate time as afine parameter can be done at all in null geodesics, and the fact is that in static spacetimes it can be done as Padmanabhan shows in his GR text:
"In the case of null geodesics in a static spacetime one can introduce another variational principle, which is a generalization of Fermat’s principle to curved spacetime. Consider all null curves connecting two events P and Q in a static spacetime. Each null curve can be described by the three functions xα(t) and will take a particular amount of coordinate time Δt to go from P to Q. We will now show that the null geodesic connecting these two events extremizes Δt. To do this, we shall change the independent variable in Eq. (4.67) from the affine parameter λ to the coordinate time t by using the relation 0=dt2 +gαβ/g00dxα dxβ. (4.75) ...The Fermat principle is now equivalent to the statement that such a gravitational field acts like a medium with a refractive index n(x)=f(x)/ |g00(x)|. In addition to the bending of light, such an effective refractive index will also lead to a time delay in the propagation of light rays. This delay, called Shapiro time-delay has been observationally verified."

Actually, that shows that coordinate time is NOT the affine parameter. Changing from the affine parameter to coordinate time as an independent (not affine) parameter allows the treatment of static gravity as an optical medium. It doesn't change that the affine parameter (for all types of geodesics) is unique up to linear function (affine parameter means the simple - parallel transport - form of geodesic equation must be satisfied using the parameter). Thus, in SC coordinates r is an affine parameter and t is not an affine parameter.

 

[TrickyDicky]

Actually, that shows that coordinate time is NOT the affine parameter. Changing from the affine parameter to coordinate time as an independent (not affine) parameter allows the treatment of static gravity as an optical medium. It doesn't change that the affine parameter (for all types of geodesics) is unique up to linear function (affine parameter means the simple - parallel transport - form of geodesic equation must be satisfied using the parameter). Thus, in SC coordinates r is an affine parameter and t is not an affine parameter.

Clearly in SC coordinates r is an affine parameter, not t. I never made a restriction to specific coordinates. Obviously a change of coordinates must be introduced for the spatial part of the metric that puts it in conformally flat form(spatial part that in the case of the statiic Schwarzschild metric is made equal to the time part in the case of null geodesics), which is perfectly valid, now you are claiming that particular coordinates are fundamental?
In general such conformal tranformations change the form of the geodesic equation and one doesn't obtain a geodesic, but the exception are null geodesics, but then the parameter is no longer affine, again static spacetimes are an exception to that and one can recover the affine parameter, in this case the coordinate t, after the transformation to a conformally flat spatial part.
Quoting once more from Padmanabhan page 160:
"...This is the same as a geodesic equation with an affine parameter t in a three-dimensional space with metric Hαβ. It follows that the null geodesics in a static spacetime can be obtained from the extremum principle for coordinate time δ ∫dt =0 , (4.79) "

 

[stevendaryl]

Clearly in SC coordinates r is an affine parameter, not t. I never made a restriction to specific coordinates. Obviously a change of coordinates must be introduced for the spatial part of the metric that puts it in conformally flat form(spatial part that in the case of the statiic Schwarzschild metric is made equal to the time part in the case of null geodesics), which is perfectly valid, now you are claiming that particular coordinates are fundamental

Nobody said that particular coordinates are fundamental. The claim is that for a specific geodesic, there are particular parametrizations that are fundamental, namely the affine ones. And an affine parametrization for an outgoing/incoming lightlike geodesic in Schwarzschild spacetime happens to be the coordinate $r$ of the Schwarzschild coordinates.

The fact that you're using $r$ as the parameter is unrelated to whether you are using Schwarzschild coordinates. It's an affine parameter for any coordinates whatsoever.

 

[TrickyDicky]

Nobody said that particular coordinates are fundamental.

I know but you and PAllen seem to be implying it wrt the Schwarzschild coordinates. See below.

The claim is that for a specific geodesic, there are particular parametrizations that are fundamental, namely the affine ones.

But I thought you understood that this is not so for null geodesics that are the subject of your OP.
Affine parametrization is not fundamental at all for null geodesics, you can parametrize them with any parameter, affine or not affine. That is a key difference wrt timelike and spacelike geodesics.
It is a particular consequence of the fact that Schwarzschild spacetime is static that one can obtain coordinate time as an affine parameter for the null geodesics. Otherwise it would not be affine.

And an affine parametrization for an outgoing/incoming lightlike geodesic in Schwarzschild spacetime happens to be the coordinate $r$ of the Schwarzschild coordinates.

I'm not disputing this at all, did you read my phrase "Clearly in SC coordinates r is an affine parameter, not t" in the previous post? The reason as explained by Bill K is that in these coordinates to express r as a function of t can't be done in closed form, you get an ugly infinite expression.

The fact that you're using $r$ as the parameter is unrelated to whether you are using Schwarzschild coordinates. It's an affine parameter for any coordinates whatsoever.

In general that would be true, but note that we are using here a parameter that hapens to be a coordinate, so for instance in coordinates that didn't use the r coordinate explicitly or in which r was not an independent variable as it is the special case with null geodesics of Schwarzschild spacetime you wouldn't be able to use the radial coordinate as affine parameter in those particular coordinates or in the case of timelike geodesics of that space.

 

[TrickyDicky]

This is the coordinate change that allows to use coordinate time as affine parameter for null geodesics: r=ρ(1+2m/2ρ)^2

The caveat as pointed out by Bill is that this change is not valid for r<2m, only r is a valid affine parameter in that case.

But then again there the spacetime is not static so it is kind of obvious.

 

[Bill_K]

Affine parametrization is not fundamental at all for null geodesics, you can parametrize them with any parameter, affine or not affine. That is a key difference wrt timelike and spacelike geodesics.

You can parametrize any curve, null or not, geodesic or not, with any parameter you choose. But for geodesics you only get the simple geodesic equation as it is usually written if you use the proper time/proper distance/affine parameter. Use of any other parametrization will produce an extra term.

It is a particular consequence of the fact that Schwarzschild spacetime is static that one can obtain coordinate time as an affine parameter for the null geodesics. Otherwise it would not be affine.

Affine parameters remain unchanged under a coordinate transformation. However under a conformal transformation, although the null geodesics themselves are preserved, the affine parameters are not.

That's what this book by Padmanabhan is doing. He's replacing Schwarzschild by a conformally related spacetime,

ds² = (1 - 2m/r) ds'²

ds'² = dt² - (1 - 2m/r)^-2 dr² - r²/(1 - 2m/r) d²Ω

In this spacetime since g₀₀ = 1, the energy integral is simply E = dt/ds, and t is indeed an affine parameter. But that's in a different spacetime, not in Schwarzschild.

 

[TrickyDicky]

You can parametrize any curve, null or not, geodesic or not, with any parameter you choose. But for geodesics you only get the simple geodesic equation as it is usually written if you use the proper time/proper distance/affine parameter. Use of any other parametrization will produce an extra term.

Exactly.
Null geodesics have the particularity that the equation with the extra term will produce also a geodesic, as all null paths are geodesics. This is not the case for timelike or spacelike geodesics, they need the nice simple equation.

Affine parameters remain unchanged under a coordinate transformation. However under a conformal transformation, although the null geodesics themselves are preserved, the affine parameters are not.

That's what this book by Padmanabhan is doing. He's replacing Schwarzschild by a conformally related spacetime,

ds² = (1 - 2m/r) ds'²

ds'² = dt² - (1 - 2m/r)^-2 dr² - r²/(1 - 2m/r) d²Ω

In this spacetime since g₀₀ = 1, the energy integral is simply E = dt/ds, and t is indeed an affine parameter. But that's in a different spacetime, not in Schwarzschild.

Not exactly.
It's not a different spacetime. It is the spatial hypersurface(wich is conformally flat) of the exterior Schwarzschild spacetime in conformally flat explicit form thru the coordinate change I gave and then making use of the fact that for the Schwarzschild null geodesic $(1-2m/r)dt^2=dr^2/(1-2m/r)-r^2dΩ^2$

 

[WannabeNewton]

Exactly.
Null geodesics have the particularity that the equation with the extra term will produce also a geodesic, as all null paths are geodesics. This is not the case for timelike or spacelike geodesics, they need the nice simple equation.

That's not what Bill is saying. A geodesic is a curve that satisfies the equation $\xi^a \nabla_a \xi^b = \alpha \xi^b$ for some scalar field $\alpha$. This is true for any geodesic whatsoever. However one can show easily that any such curve can be reparametrized so as to satisfy $\xi^a \nabla_a \xi^b = 0$. This is called an affine parametrization for geodesics.

Back to the original question: as I have tried to state, any search for a general physical meaning, in arbitrary curved space-times, for arbitrary affine parameter will be futile, and for obvious reasons from elementary differential geometry. Arbitrary curve parameters are not meant to be physical and in general have no physical meaning whatsoever. They are bookkeeping parameters that is all. Some special cases allow for physical meanings, such as the arc-length parametrization (which is where proper time comes from) but for null geodesics such a parametrization does not exist.

 

[Bill_K]

Sorry, TrickyDicky, I stand by my remarks.

 

[WannabeNewton]

It's not a different spacetime. It is the spatial hypersurface(wich is conformally flat) of the exterior Schwarzschild spacetime in conformally flat explicit form thru the coordinate change I gave and then making use of the fact that for the Schwarzschild null geodesic $(1-2m/r)dt^2=dr^2/(1-2m/r)-r^2dΩ^2$

No it's a conformally related space-time.

 

[TrickyDicky]

That's not what Bill is saying.

I know that's what I'm saying.:tongue:

A geodesic is a curve that satisfies the equation $\xi^a \nabla_a \xi^b = \alpha \xi^b$ for some scalar field $\alpha$. This is true for any geodesic whatsoever. However one can show easily that any such curve can be reparametrized so as to satisfy $\xi^a \nabla_a \xi^b = 0$. This is called an affine parametrization for geodesics.

This is understood, I was adding to this my reason to stress the case of null geodesics.