Derivations

Derivation of the photon coordinate path equation.

Start with the radial only Schwarzschild metric.

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

For a photon the proper time dS is zero and the coordinate velocity dr/dt of a photon is easily solved to give:

The above equation then needs to be integrated to obtain a result in terms of r and t. This is most easily achieved by inverting the equation to obtain dt/dr and then integrating with respect to r to obtain:

$\int_{r_1}^{r_2}\left(\frac{dt}{dr}\right){dr}=\int_{r_1}^{r_2}c \left( 1-\frac{2m}{rc^2}\right)^{-1} {dr} = \left[(rc^2+2m \,\,ln(2m-rc^2))/c^3\right]_{r_1}^{r_2}$

Inserting the values r1 and r2 for r gives:

$t=\frac{(r_{_2}-r_{_1})}{c} + \frac{2m}{c^3} \,ln\left(\frac{2m-r_{_2}c^2}{2m-r_{_1}c^2}\right)$

The log of a negative number is imaginary and the reader should satisfy themselves that a real solution is only obtained when r1 and r2 are both greater than 2m or both are less than 2m. Therefore there is no continuous path across the event horizon for a photon. A photon can approach the event horizon from above or below but it can not cross it.

Most text books get around this problem by inserting an absolute function inside the log function to force the result to be real and prove that the photon crosses the event horizon. There is no mathematical justification for that and in fact it is mathematically and intellectually unethical and unscientific to do that. You can not change the rules of maths to give the result that you are biased towards.

Derivation of the particle coordinate path equation.

The derivation of the particle path is somewhat more involved and the reader is referred to the mathpages derivation by Kevin Brown. As far as I can tell the derivation is correct except that Kevin Brown succumbs to the usual temptation to insert an ad-hoc absolute function so that the result agree with the accepted traditional physical interpretation rather than let the maths speak for itself and question the accepted physical interpretation. These are the final corrected steps shown below.

Starting with:

$t = \sqrt{{R \over 2m} -1}\left[\left( {R \over 2} +2m \right)\alpha + \left(R \over 2\right) sin(\alpha)}\right] - 4m\sqrt{-1}\,\,arctan\left(cos(\alpha)-1 \over sin(\alpha)\sqrt{1-R/2m} \right)$

Substituting

the equation becomes:

$t = \left( {R \over 2} +2m \right)Q\alpha + \left(R \over 2\right)Q \,sin(\alpha)}\right] -4m\sqrt{-1}\,\,arctan\left(cos(\alpha)-1 \over sin(\alpha)\sqrt{-1}\,Q \right)$

Substituting the identity

we get

$t = \left( {R \over 2} +2m \right)Q\alpha + \left(R \over 2\right)Q \,sin(\alpha)}\right] -4m\sqrt{-1}\,\,arctan\left(-tan(\alpha/2) \over \sqrt{-1}\,Q \right)$

Now the final substitution of

gives:

$t = \left( {R \over 2} +2m \right)Q\alpha + \left(R \over 2\right)Q \,sin(\alpha)}\right] -4m \, ln\left(Q+tan(\alpha/2) \over Q-tan(\alpha/2) \right)$

and not

$t = \left( {R \over 2} +2m \right)Q\alpha + \left(R \over 2\right)Q \,sin(\alpha)}\right] -4m \, ln\left(\left| Q+tan(\alpha/2) \over Q-tan(\alpha/2)\right|\right)$

as claimed by Kevin Brown. When I first pointed this out to K. Brown he removed the ad-hoc absolute function in the last term, but unfortunately he put it back in again later when he realised it did not agree with the accepted physical interpretation.

The alpha variable that appears in the particle equation is a parametric variable. It can be removed to give a normal Cartesian equation by noting that mathpages gives the radial part of the proper motion of a particle as r=R(1+cos(alpha))/2 which is easily solved to give alpha = acos(2r/R-1) which is the form used in the applets. Later it is shown that using the Cartesian version offers valuable insight into the nature of time itself that is obscured by using the parametric equations.

Update 21^st Dec 2008

Since I posted the above comments about the mathpages derivation, Kevin Brown has updated his page to try and justify the insertion of the absolute function into the equation. His argument is basically that the equation is discontinuous across the event horizon and therefore the constant of integration is allowed to be different above and below the event horizon. His argument is correct to a point but fails in its application. Below the event horizon the logarithmic term on the end has a negative argument and becomes imaginary making the solution for t a complex value and below the event horizon the equation has this form:

$t = \left( {R \over 2} +2m \right)Q\alpha + \left(R \over 2\right)Q \,sin(\alpha)}\right] - 4m \, ln\left(\left| Q+tan(\alpha/2) \over Q-tan(\alpha/2)\right|\right)+ {\color{red} i \pi }$

At this point Kevin Brown chooses to add a constant of integration of (-i*pi) and to interpret the coordinate time of the motion in terms of only the real part of the complex equation. He is free to do that, but when the apogee is below the event horizon (R < 2m) the coordinate motion is in total contradiction to the motion in proper time. The former shows the particle falling to the event horizon and all time values for r<R have no real part, while the latter shows the particle falling to the central singularity ad all proper time values for r>R have no real part. By choosing to add a constant of (-i*pi) Kevin Brown (and most text books) is choosing to accept a singularity of infinite mass density as a physically real and he is also choosing to accept that light travels backwards in time below the event horizon.

On the other hand, I am free to choose not to add a constant of integration and by doing so, the contradiction of coordinate motion and proper motion is removed, the central singularity of infinite density is removed and we are not required to accept that light travels backwards in time below the event horizon. In other words all the paradoxes of the conventional interpretation are not an inevitable conclusion of General Relativity. I have shown that there is an equally mathematically valid interpretation that removes all the contradictions and paradoxes.

It seems Kevin Brown has not investigated the implications of setting the apogee to values of (R <2m) in his own equations as I have. I am sure that if he does, he will find it a revelation. Kevin Brown is clearly a very intelligent person and very knowledgeable in geometry, calculus and General Relativity (I suspect he is a professor) and I would love for him to investigate my claims stated in this website with an open mind. I have known for a long time that ‘something’ is wrong with the conventional interpretation of black holes, but I would not have been able to demonstrate that mathematically without the equations derived by Kevin Brown. In that sense Kevin Brown deserves 90% of the credit for my work here and we differ only in the final physical interpretation of the equations.

I have no direct contact with Kevin Brown, but he has been updating his website in response to my questions and objections here and elsewhere on the internet. The evolution of his webpage (Radial paths in a Spherically Symmetric Field) is documented in this list with approximate dates. (Sometimes it is a couple of days before I notice the webpage has changed).

Radial paths in a Spherically Symmetric Field 6thOct2008

Radial paths in a Spherically Symmetric Field 21stNov2008

Radial paths in a Spherically Symmetric Field 23rdNov2008

Radial paths in a Spherically Symmetric Field 27thNov2008

Radial paths in a Spherically Symmetric Field 20thDec2008

Until further notice I can be contacted at my physicsforums blog here or at the email address given in the introduction.

Please do not post questions about this website in the public part of the physicsforums, as posting anything relating to a website that questions the mainstream view can earn you an instant ban. Even posting something like “I don’t think this website is right” and posting a link to this website can get you banned. You have been warned.

Back to the index.