Not sure about this coordinate definition

In summary, the Eddington-Finkelstein coordinates were derived by starting with the worldline of a radially ingoing photon and integrating the dt/dr with negative sign. C is a continuous parameter that selects which geodesic out of that family we are talking about. The equation defines a whole family of null geodesics. v is the advanced time coordinate that is equal to the worldline's t_0.
  • #1
TrickyDicky
3,507
27
In the derivation of the Eddington-Finkelstein coordinates in Schwarzschild spacetime we started with the worldline of a radially ingoing photon:
[tex]ct=-r-2mln(\frac{r}{2m}-1)+C[/tex]
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.
[tex]v=ct+r+2mln(\frac{r}{2m}-1)[/tex]
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?
 
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  • #2
TrickyDicky said:
In the derivation of the Eddington-Finkelstein coordinates in Schwarzschild spacetime we started with the worldline of a radially ingoing photon:
[tex]ct=-r-2m \ln (\frac{r}{2m}-1)+C[/tex]
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.
 
  • #3
Ben Niehoff said:
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?
 
  • #4
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:

[tex]t - t_0 =-r-2m \ln (\frac{r}{2m}-1)[/tex]

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 [itex]v = t_0[/itex]. Simple.
 
  • #5
Ben Niehoff said:
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:

[tex]t - t_0 =-r-2m \ln (\frac{r}{2m}-1)[/tex]

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 [itex]v = t_0[/itex]. Simple.

Ok, it's clear now, thanks.
 

FAQ: Not sure about this coordinate definition

1. What is the definition of coordinates?

Coordinates are a set of numbers that are used to specify the position of a point or object in a given space. They typically consist of two or three values that indicate the location in relation to a reference point or origin.

2. How are coordinates represented?

Coordinates can be represented in various ways, but the most common methods are Cartesian coordinates, which use a combination of x, y, and z values, and polar coordinates, which use a distance and angle to specify location.

3. What is the purpose of using coordinates?

Coordinates are used to accurately describe the position of objects or points in a given space. This is important in fields such as mathematics, geography, and physics, where precise location information is necessary for calculations and analysis.

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5. How do I convert between different coordinate systems?

To convert between different coordinate systems, you can use mathematical formulas or specialized software. It is important to understand the properties and relationships of each system in order to accurately convert between them.

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