Not sure about this coordinate definition

[TrickyDicky]

In the derivation of the Eddington-Finkelstein coordinates in Schwarzschild spacetime we started with the worldline of a radially ingoing photon:
$$ct=-r-2mln(\frac{r}{2m}-1)+C$$
where C is a constant of integration since we got this from integrating the dt/dr with negative
sign from the Schwarzschild radially moving photon.
The next step to introduce the new Finkelstein coordinate (wich would be the advanced time v) is to use the integration constant given in the photon worldline to define this new coordinate that allows us to say that 2m=r is not a real singularity.
$$v=ct+r+2mln(\frac{r}{2m}-1)$$
What I don't see clearly in this step is how come we use a constant to define a coordinate, I would have thought a coordinate is not usually a constant, it can be momentarily for certain purposes like when we hold one of the coordinates fixed to see what happens, like examining a constant time hypersurface, or when we take advantage of some symmetry like spherical symmetry to hold constant phi and theta coordinates. But I just don't see why would we want to keep the advanced time E-F coordinate constant.
Anyone has a clue about this?

 

[Ben Niehoff]

In the derivation of the Eddington-Finkelstein coordinates in Schwarzschild spacetime we started with the worldline of a radially ingoing photon:
$$ct=-r-2m \ln (\frac{r}{2m}-1)+C$$
where C is a constant of integration since we got this from integrating the dt/dr with negative
sign from the Schwarzschild radially moving photon.

This equation defines a whole family of null geodesics. C is a continuous parameter that selects which geodesic out of that family we are talking about. So we can define a coordinate system in which C is one of the coordinates.

 

[TrickyDicky]

This equation defines a whole family of null geodesics. C is a continuous parameter that selects which geodesic out of that family we are talking about. So we can define a coordinate system in which C is one of the coordinates.

Thanks Ben, is there not a redundancy in the parametrization of this space by introducing this continuous parameter? perhaps parametrization invariance here poses a problem?

 

[Ben Niehoff]

Not sure what you're talking about. You seem to be reading way too much into things and confusing yourself.

It's really quite simple. Let me re-write the geodesic like so:

$$t - t_0 =-r-2m \ln (\frac{r}{2m}-1)$$

So, all we are doing is choosing a different t_0. All these geodesics follow the same path in 3-space, but start at different times. That's all there is to it.

Now we just define $v = t_0$. Simple.

 

[TrickyDicky]

Not sure what you're talking about. You seem to be reading way too much into things and confusing yourself.

It's really quite simple. Let me re-write the geodesic like so:

$$t - t_0 =-r-2m \ln (\frac{r}{2m}-1)$$

So, all we are doing is choosing a different t_0. All these geodesics follow the same path in 3-space, but start at different times. That's all there is to it.

Now we just define $v = t_0$. Simple.

Ok, it's clear now, thanks.