The flaws in Eddington-Finkelstein coordinates

The flaws in Eddington-Finkelstein coordinates.

In the preamble to the derivation of the Eddington-Finkelstein coordinates on page 353 of this text book the speed of light (dr/dt) in Schwarzschild coordinates is given in Equation 13.2 as :

dr = (1-2*m/r) dt

This can be rearranged to:

dt/dr = 1/(1-2*m/r) which is easily integrated with respect to r to obtain:

t = (+/-) (r+2*m*ln(r-2*m) +const)

and not

t = (+/-) (r+2*m*ln(abs(r-2*m)) +const) as claimed in the book.

This can be confirmed by copying and pasting r+2*m*ln(r-2*m) into this online mathematica differentiator and seeing that result differs from differentiating r+2*m*ln(-r+2*m). So why the ad-hoc and erroneous introduction of the absolute function into the equation? The reason is that when r is less than 2m the log function returns imaginary results. This does not agree with the result published in most text books that a photon passes through the event horizon. Obviously the authors of this book ( and I am not singling out any book in particular – all text books on the subject commit the same crime) decided that if the mathematical result does not agree with the expected result then change the rules of maths!

So far we have established that in Schwarzschild coordinates at least, a photon stops at the event horizon and does not pass through.

In the text book and even in most of this website the speed of light c is taken to be one and c is left out of the equations for simplicity. This is perfectly reasonable most of the time, but when studying light paths in detail and especially when c is not in its squared form, the sign of c becomes important and it is wise to reintroduce c as an explicit variable in the equations. I will do that from here on.

With c explicitly stated, the purely radial Schwarzschild metric given as equation 13.1 in the book is:

$c^2 dS^2 = \left(1-\frac{2m}{rc^2}\right)^{-1}dr^2-\left(1-\frac{2m}{rc^2}\right)c^2dt^2 c^2 ds^2 = (1-2*m/r)^(-1)dr^2 - (1-2*m/r)*c^2*dt^2$

The book then introduces on page 355, a new time parameter called t* defined as:

$t^{*} = t + (2m/c^3)}\, ln\left|rc^2-2m\right|}$

and then performs implicit differentiation of t with respect to r to obtain:

$dt = dt^{*} -2m\frac{ dr}{c(rc^2-2m )}$

which is correct if the absolute function in the previous equation is ignored, but since the absolute function was introduced with no mathematically justification, ignoring it is the best option.

Now this value for dt is substituted into the original purely radial Schwarzschild solution to give:

$c^2 dS^2 = \left(\frac{2m}{rc^2}+1\right)dr^2 +\left(\frac{4m}{rc}\left)drdt^* -\left(1-\frac{2m}{rc^2}\right)c^2dt^{*2}$

For a photon dS=0 and the equation takes on a quadratic form which has two solutions for dr/dt* which are:

$\frac{dr}{dt^*} = -c$ and $\frac{dr}{dt^*} = c \frac{(rc^2-2m)}{(rc^2+2m)}$

Now it becomes clear that these Eddington-Finkelstein coordinates are ambiguous to say the least. Are we to take the first solution and assume a photon in these coordinates is going in the opposite direction to the same photon observer by a Schwarzschild observer? What sort of velocity would an observer have to have to make a photon appear to reverse direction? So although it is never stated whose point of view the measurements are made from in EF coordinates it seems that the implication is that it is the point of view of an observer going faster than the local speed of light which makes the point of view questionable. If we take the second solution then we see that at the event horizon when r=2m the velocity of a photon is zero. This does not support the view that a photon passes smoothly through the event horizon because clearly the photon stops at the event horizon in these coordinates. How long does it stop for before it continues on its alleged inevitable journey to the central singularity?

In the diagram in the book the first solution of dr/dt* = -c has been chosen to represent the ingoing photons which in Schwarzschild coordinates is an outgoing photon. The second solution has been chosen to depict the motion of an outgoing photon in these coordinates but clearly the choice is arbitrary and could be reversed simply by reversing the sign of c used in the equations. It is not safe to reach any physical conclusions from these particular coordinates.

Now it should be noted that Eddington-Finkelstein coordinates have several variants. All claim to show that photons pass smoothly through the event horizon and below the event horizon photons only move towards the central singularity. On that basis all Eddington-Finkelstein variants are flawed. Since Eddington-Finkelstein coordinates are popular in text books and used as the basis for many studies and interpretations of the physical events the implication is that almost everything you may have learned about the physical interpretation of events below the event horizon of a black hole, from a text book are wrong.

Another EF variant is often stated in this form:

dS^2 = -(1-2m/r)dt*^2 + 2drdt*

See this Wikipedia article.

When the derivation is looked at in detail using c explicitly rather than assuming c=1 the actual solution is

-c^2dS^2 = -c^2 f dt*^2 + 2c^2 drdt* + (1-c^2)dr^2 / f where f = (1-2m/(r c^2))

because the explicit c does not allow the dr^2 term to cancel out. Once again when dS is set to zero, the equation takes on a quadratic form, the solution of which is

dr/dt* = cf/(c +/- 1)

If c is taken to be +1 then the solution is f/2 or infinity.

If c is taken to be -1 then the solution is f/2 or infinity.

In other words it does not matter what direction the photon has in Schwarzschild coordinates, the photon can only go in one direction in these coordinate whether it is above the event horizon or not.

To avoid the infinity result a non unity value of has to be chosen for c. For example let us choose units such that c=2. The solution then becomes

dr/dt* = 2(1-m/(2r))/3 or dr/dt* = 2(1-m/(2r))

and the same pair of solutions is the same if c=-2 is chosen. Once again the choice of solution is arbitrary and inconclusive because of the quadratic nature of the equation. It can also be noted that with a choice of c equal to plus or minus c that the photon velocity goes to zero when r=m/2 so the event horizon has been moved to a different location and that even so that the photon comes to a stop at this shifted event horizon before it arrives at the central singularity. Clearly there are serious problems with this metric.

In Schwarzschild coordinates the expression for the coordinate velocity of light with c explicitly stated is:

dr/dt = c(1-2m/(rc^2)) See derivation here

and it is easy to see that that the direction is determined by the variable c in those coordinates.

Now if we take another look at this expression given earlier:

-c^2dS^2 = -c^2 f dt*^2 + 2c^2 drdt* + (1-c^2)dr^2 / f where f = (1-2m/(r c^2))

and take a value for c of +/-1 then the last term is zero for either sign of c so it is reasonable to write the equation as:

-c^2dS^2 = -c^2 f dt*^2 + 2c^2 drdt*

By setting dS to zero dividing both sides by dt*^2 the solution

dr/dt* = c^2 f / (2c^2) = 2/f is readily obtained.

Note that there is no sign for direction in the equation dr/dt* = f/2 for light in Eddington-Finkelstein coordinates, so this metric only has a one way direction for light above and below the event horizon which is clearly not correct.

Obviously it would be time consuming to analyse all Eddington-Finkelstein coordinate variations so that task will have to go on my ‘to do’ list. If an interested reader knows of a clear derivation of an EF variant and wants to know where the flaws are, please send the details in ;)

Until further notice I can be contacted at my physicsforums blog here.

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