# Affine parametrization for null geodesic?

Staff Emeritus

## Main Question or Discussion Point

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, $g_{\mu \nu}U^\mu U^\nu = 0$)?

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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.)

Staff Emeritus
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.

Staff Emeritus
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.

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?

Staff Emeritus
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.

Staff Emeritus
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.

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.

Staff Emeritus
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
2019 Award
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.

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.

Staff Emeritus
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.

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.

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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 wich 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.

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.

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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.

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.]

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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/g00dxα 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