This is a follow on from a previous thread about issues I had with regular Eddington-Finkelstein coordinates. First I show a brief derivation of the advanced EF coordinates which appears to have less issues than the regular EF coordinates and then discuss what I think appear to be the remaining issues.(adsbygoogle = window.adsbygoogle || []).push({});

The purely radial Schwarzschild metric is given by:

[tex]dS^2 = \left(1-\frac{2m}{rc^2}\right) dt^2 c^2 -\ \frac{dr^2}{ (1-\frac{2m}{rc^2})} [/tex]

The relationship of the Schwarzschild time coordinate (t) to the advanced Eddington Finkelstein time coordinate (T) is defined by:

[tex]dt\ =\ dT -\frac{2m}{c(rc^2-2m)}dr [/tex]

Substituting this equation for dt into the radial Schwarzschild metric and simplifying gives:

[tex]dS^2 = \left(1-\frac{2m}{rc^2}\right) dT^2 c^2

- \frac{4m\ dT\ dr \ c}{r}

-\ \left(1+\frac{2m}{rc^2}\right)dr^2 [/tex]

For a photon, dS is is set to zero and solving for dr/dT gives:

[tex]\frac{dr}{dT} = -c[/tex]

and

[tex]\frac{dr}{dT} = c \frac{ \left(1-\frac{2m}{rc^2} \right)}{\left(1+\frac{2m}{rc^2} \right)}[/tex]

The above results show that the equation for an ingoing light ray in advanced EF coordinates behaves nicely at the event horizon with no infinities. However it can also be noted that the coordinate velocity of a outgoing photon is zero at the event horizon in these coordinates and arrives at the event horizon in infinite coordinate time, just as it is in Schwarzschild coordinates.

Integrating dT with respect to r for the outgoing photon gives:

[tex]T = \frac{r}{c} + \frac{4m\ \log(2m - rc^2)}{c^3} [/tex]

Plotting T versus r gives:

https://www.physicsforums.com/blog_attachment.php?attachmentid=114&d=1259383629 [Broken]

which is identical to the ingoing regular EF coordinates after the r axis has been tilted. Note that these coordinates are still not perfect, as there is some uncertainty about whether or not the path of the outgoing photon is continuous across the event horizon at T = -infinity.

Reversing the time coordinate gives the outgoing advanced EF coordinates as plotted below.

https://www.physicsforums.com/blog_attachment.php?attachmentid=115&d=1259383629 [Broken]

The outgoing coordinates are identical to the regular EF coordinates after the r axis has been tilted in the opposite direction to the r axis of the ingoing regular EF coordinates.

The outgoing coordinates (as I understand it) represent a white hole. One issue that I have here, is that light and particles can fall to the event horizon of the white hole in infinite coordinate time and this translates to particles falling to the white hole event horizon in finite proper time. Is that supposed to happen?

As mentioned above, ingoing and outgoing Eddington-Finkelstein coordinates still have uncertainties about the continuity of null paths as they cross the event horizons at T = plus or minus infinity, where the velocity of the photon goes to zero exactly at the event horizon in these coordinates.

I would like to introduce a new coordinate transformation that appears to remove this uncertainty.

First I define a new time parameter T related to the Schwarzschild time parameter t by the following relationship:

[tex]dt\ =\frac{dT} {\left(1+\frac{2m}{rc^2}\right) \left(1-\frac{2m}{rc^2} \right)} [/tex]

Substituting this new definition of dt into the radial Schwarzschild metric and simplifying gives:

[tex]dS =\frac{c^2 dT^2} {\left(1+\frac{2m}{rc^2}\right)^2 \left(1-\frac{2m}{rc^2} \right)} - \frac{dr^2} {\left(1-\frac{2m}{rc^2} \right)} [/tex]

For a photon dS is set to zero and solving for dr/dT gives:

[tex]\frac{dr}{dT} = \pm \frac{ c}{\left(1+\frac{2m}{rc^2}\right) } [/tex]

Integrating dT with respect to r gives:

[tex]T = \pm \left( \frac{r}{c} +\frac{2m \log(r)}{c^3} \right) [/tex]

which is the light paths in terms of T and r which can be plotted to give the graph below:

https://www.physicsforums.com/blog_attachment.php?attachmentid=116&d=1259383629 [Broken]

I have called the new coordinates "fast coordinates" as they are sort of the opposite of Finkelstein's "tortoise coordinates". The red curve in the graph represents a typical ingoing light path and the green curve represents a typical outgoing light path.

This new metric has the advantage that light paths are smooth and continuous across the event horizon for both ingoing AND outgoing photons. In these new coordinates, the "singularity" at the event horizon has been transformed away, perhaps too successfully, because the statement that there is "nothing special about the event horizon" now means that is is not even what Finkelstein called a "one way membrane" in these new coordinates. The new coordinates also remove the requirement to have ingoing and outgoing versions of the metric. White holes no longer exist in these new coordinates and black holes no longer have the properties we are used to. Clearly something appears to be wrong. It should be impossible to arrive at different physical conclusions, just by using different transformations, because General Relativity is supposed to be a diffeomorphism invariant theory. Can anybody spot where I made the error in the transformation and what rules govern what are allowable transformations?

I have other “issues”, but I will come back to them after this issue has been addressed.

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# Issues with advanced Eddington-Finkelstein coordinates

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