Kerr Metric Time Dilation Formula: Deriving Absolute Form

[Dukon]

TL;DR Summary

convert a differential version to absolute version of time dilation formula

Just as the time dilation formula for the Schwarzschild metric in terms of the position $r$ away from center of mass for a gravitational body and the Schwarzschild radius $r_s = {2GM}/{c^2}$ is given by
$$ \tau = t \sqrt{1 - \frac{r_s}{r} } $$
so I'd like to know the corresponding absolute (non-differential) form for the Kerr metric but all I could find for the Kerr metric time dilation formula was some differential ratio subtracted from unity. In terms of the modified distance $\rho = f(r)$ which reduces to $r$ if $J=0$ the Kerr time dilation is apparently given by (if this can be trusted)

$$ \eqalign{
\rho^2 &= r^2 + \left( \frac{J}{Mc} \right)^2 \cos^2 \theta \cr

1 - \left( \frac{{\rm d}\tau}{{\rm d}t} \right)^2 &= \frac{2 GM r}{c^2 \rho^2}
}
$$How can I a) get rid of the subtraction from unity, and b) not have it in the differential form but rather just the absolute form like above for the Schwarzschild metric: $\tau$, time on a moving frame relative to a reference frame on which is the reference time $t$?

If anyone has derived these other absolute time dilation formulae for the Kerr-Newman and Reissner-Nordstrom cases as well please do share.

 

[Orodruin]

Since everything is constant for a stationary observer apart from the time coordinate $t$, going from the differential form to the integrated form should be trivial - just replace the differentials by the integrated quantities. Everything here is just a simple matter of doing that and solving for $\tau$.

 

[Dukon]

Thank you

I was just now looking how to delete my post because after I posted it, I was able to solve myself both a) and b) of my request. Your comment is helpful confirmation that what I did was correct. Glad you commented, thanks.

The Kerr time dilation in the form I wanted it to be can be written now as
$$ \tau = t \sqrt{1 - \frac{r_s r}{\rho^2} } $$

I think I should still delete this post, and just ask for the starting points for the Kerr-Newman and Reissner-Nordstrom cases

 

[PeterDonis]

I was just now looking how to delete my post because after I posted it, I was able to solve myself both a) and b) of my request.

That doesn't mean you either need to or should delete your post. Other readers of PF might have the same question and your answer will benefit them.

the starting points for the Kerr-Newman and Reissner-Nordstrom cases

What starting points do you mean?

 

[Dukon]

Thank you for your comments on this

By starting point I mean the Kerr expressions in my first post. To me, since that was not in a simple absolute form of $\tau$ on left and $t$ on the right, I took the whole Kerr mess I found somewhere else as my Kerr starting point. After doing the algebra it no longer looks like a mess and in fact in the end both appear as equivalent but different ways of representing the same physics.

I am not GR conversant enough to derive time dilation formulae from any given metrics, but I would like to do algebra upon them and use them in my own calculations just to see what difference Q and J make for calculations involving planets in our Solar System. In short, I would like to use the results which I cannot myself derive from GR metrics. So I assume these have already been worked out by those conversant with Kerr-Newman and Reissner-Nordstrom metrics but never seem to find them already presented amidst the myriad other points being made in the literature about these other two metrics.

What would be nice to have is just to have all 4 time dilation formulae side by side all in the same what I call absolute format, just to at a glance see the differences as now can be done above between Schwarzschild and Kerr.

J for each planet should be easily deducible from lists of their hourly or daily rotations, but where I wonder would I find the Q for the planets. Does anyone have information on Q measurements which have been done for the planets?

 

[PeterDonis]

I am not GR conversant enough to derive time dilation formulae from any given metrics

How did you get the answer you gave in post #3?

 

[PeterDonis]

just to see what difference Q and J make for calculations involving planets in our Solar System

They don't make any difference. Q is effectively zero for any astronomical object (since any nonzero charge on an object that large will attract opposite charges to the object that will neutralize it) and J for all objects is too small to have any significant effect on the spacetime geometry on the scale of the solar system (i.e., things like the orbits of planets).

For a case where J has a detectable effect, you might look up Gravity Probe B, which measured one of the (small but detectable) effects that the Earth's J has on the spacetime geometry in its immediate vicinity.

 

[PeterDonis]

all 4 time dilation formulae

Where are you getting 4 from?

 

[Dukon]

How did you get the answer you gave in post #3?

Using the expression for the Schwarzschild radius $r_s$ and the "starting point" Kerr expression both from post #1, then the Kerr expression in terms of $r_s$, the position $r$ and Angular momentum modified position $\rho$ reduces to $$1 - \left( \frac{{\rm d}\tau}{{\rm d}t} \right)^2 = \frac{2 GM r}{c^2 \rho^2} = \frac{r_s r}{\rho^2}$$ Next, subtract both sides from their own unity, take the square root of both sides and the result is given in post #3. Orodruin's point in #2 is that nothing is time dependent (everything is a constant) so integration converts the differentials into absolutes, and the subtraction from unity was actually needed after all but under the square root -- so there was nothing ill about it being on the left hand side in the first place. In fact, moving it to the left hand side is strategic for isolating the right hand side "business term" by itself, the only interesting part of Time Dilation, namely whatever this quantity is that is being subtracted from unity under the square root in familiar time dilation.

Where are you getting 4 from?

The 4 comes from the 2x2 matrix of exact solutions to the Einstein Field Equations of 2 columns being labeled by $J=0$ and $J\ne 0$ and the 2 rows being labelled by $Q=0$ and $Q\ne 0$.

They don't make any difference.

That the differences might not be practical is not my direct interest; I am interested rather in just what are the literal exact differences no matter how small they might be in practical human terms. If they are hundreds of orders of magnitude too small to be detected that is ok by me. I just want to see what the calculation gives no matter what Qs and Js are entered, even if these are many orders of magnitude smaller than unity. I just want to be armed with the exact expressions, so I can avoid just being satisfied with some general statement that they are small or negligible. I want to know exactly how negligible.

Armed with exact expressions, it could be a matter of later concern how or whether they are practical. Something I think is fun to do so armed is to dictate whatever values Q and J must have in order to become practical in human terms. The learning, seeing and knowing just how astronomical those values are is part of the education for the extremities from human scale reality GR actually is; in short, I think this sort of education is useful.

 

[PeterDonis]

the "starting point" Kerr expression both from post #1

Which comes from the Kerr metric. Specifically, from the metric coefficient $g_{00}$ in the Kerr metric.

The 4 comes from the 2x2 matrix of exact solutions to the Einstein Field Equations of 2 columns being labeled by $J=0$ and $J\ne 0$ and the 2 rows being labelled by $Q=0$ and $Q\ne 0$.

These aren't 4 different solutions. They're one solution, the Kerr-Newman solution, which contains all 4 of your cases. So all you need to do is look at the $g_{00}$ metric coefficient in the Kerr-Newman solution and apply the same procedure you have already used for the Kerr solution.

 

[Dukon]

Absolutely fantastic to learn. Thank you.

However, as I am not GR conversant enough I drown when viewing the myriads of other points made in the literature about any metric, that for me to find the $g_{00}$ metric coefficient would take hours and days still not knowing where it is. Even on the Wikipedia page for Kerr-Newman metric. Any chance you can say write it down in here to see it here as opposed to elsewhere?

 

[Ibix]

$g_{00}$ is the coefficient of $dt^2$ in the line element, which is the first expression in the Boyer-Lindquist coordinates section of Wikipedia's Kerr-Newman page. Expand that, then set $dr=d\theta=d\phi=0$ and you'll get an expression that looks like $c^2d\tau^2=(\mathrm{something})c^2dt^2$. The "something" is $g_{00}$, the time dilation factor (edit: the square of the time dilation factor actually - bed time, I think...).

Sanity check - setting $Q=J=0$ should recover the Schwarzschild expression, and setting $Q=0$ should recover the Kerr expression.

 

[Dukon]

Thank you very much for sharing the steps I would not have known to take from the place on that page I would never have known to stop to find this. I will post my final result of what I call the absolute form of time dilation which reduces in the $J=Q=0$ limits to the Schwarzschild time dilation expression in post #1.

Your help is very much appreciated

 

[Dukon]

The Time Dilation formula for the Kerr-Newman metric, in terms of a number of length parameters for a position $(r,\theta)$ away from center of mass and polar axis respectively of a gravitational body of mass $M$, polar angular momentum $J$ and electrical charge $Q$, is independent of azimuthal angle $\phi$ and expressed as: $$
\eqalign{
g_{00} &= \left[ \frac{\Delta}{\rho^2} - \frac{a^2 \sin^2 \theta}{\rho^2} \right] \\
a &= \frac{J}{Mc} \cr
\rho^2 &= r^2 + a^2 \sin^2 \theta \cr
\Delta &= r^2 + {r_Q}^2 - r_s r + a^2 \cr
\Delta - a^2 \sin^2 \theta &= r^2 + {r_Q}^2 - r_s r + a^2 \cos^2 \theta \\
{r_Q}^2 &= k \ \frac{Q^2 G}{c^4} \ \ {\rm where} \ \ k = \frac{1}{4 \pi \epsilon_0} \ \ {\rm from \ Electrostatics} \\[2mm]
\tau &= t \ \sqrt{1 - \frac{r_s r + {r_Q}^2}{\rho^2} } \hspace{1in} {\rm Kerr-Newman} \ J\ne 0 \ Q\ne 0.
} $$ And so all four together for convenience are $$
\eqalign{
\tau &= t \sqrt{1 - \frac{r_s}{r} } \hspace{1.7in} \ J=0 \ Q=0 \ \ {\rm Schwarzschild} \cr
\tau &= t \sqrt{1 - \frac{r_s r}{\rho^2} } \hspace{1.6in} \ J\ne 0 \ Q = 0 \ \ {\rm Kerr} \cr
\tau &= t \sqrt{1 - \frac{r_s}{r} + \frac{{r_Q}^2}{r^2} } \hspace{1in} \ J=0 \ Q\ne 0 \ \ {\rm Reissner-Nordstrom} \cr
\tau &= t \sqrt{1 - \frac{r_s r + {r_Q}^2}{\rho^2} } \hspace{1in} \ J\ne 0 \ Q\ne 0 \ \ {\rm Kerr-Newman}.
}.
$$ Many thanks are extended to all responders to the initial post since the GR pages could never have been navigated without expert PF help! Please correct if errors persist.

Lastly, does the azimuthal independence reflect a symmetry with general implications for GR and does this have a special name to use as search keyword besides "azimuthal symmetry in GR"?

 

[PeterDonis]

does the azimuthal independence reflect a symmetry with general implications for GR

Yes, it's called axial symmetry, and a spacetime with this symmetry is called axially symmetric or (more commonly) axisymmetric.

Note that the metric is also independent of the time coordinate $t$, which indicates another symmetry; this symmetry is called time translation symmetry, and a spacetime with this property is called stationary.

For the $J = 0$ case, i.e., Schwarzschild or Reissner-Nordstrom, the time translation symmetry has an additional property called "hypersurface orthogonal", and the spacetime is said to be static. Also, these spacetimes are spherically symmetric instead of just axisymmetric. There is actually a theorem called Birkhoff's Theorem that says that any spherically symmetric vacuum spacetime (i.e., $Q = 0$) must be isometric to Schwarzschild spacetime, and a generalization of this theorem says that any spherically symmetric electrovacuum spacetime (i.e., the only stress-energy present is from an electromagnetic field, so we can have $Q \neq 0$, but no other matter or energy is present) must be isometric to Reissner-Nordstrom.

For the $J \neq 0$ case, however, there is no analogue to Birkhoff's theorem or its generalization. This means that, while we can say that the vacuum gravitational field around a spherically symmetric (non-rotating) star or planet is exactly described by the Schwarzschild (or Reissner-Nordstrom) geometry, we cannot say that the vacuum gravitational field around a rotating (i.e., only axisymmetric, not spherically symmetric) planet or star is exactly described by the Kerr (or Kerr-Newman) geometry. It took some time after the initial discovery of the Kerr solution for physicists to realize that it described, not a rotating star or planet, but a rotating black hole.

 

[Dukon]

$g_{00}$ is the coefficient of $dt^2$ in the line element ... (edit: the square of the time dilation factor actually - bed time, I think...).

Actually, I think it is ok not to square it. I left $g_{00}$ as the $(something)$ as it is written top of my post #14 without taking it as $(something)^2$. Leaving it as I did, when push came to shove, I had to take the square root of both sides to get rid of the squares on the $\tau,t$ differentials, and that naturally gave the familiar Lorentz factor looking square root. Had I squared as you suggest, then there would be no square root in my final expression for $\tau = t \sqrt{...}$, but since that appears to be needed I don't think squaring it was needed after all.

 

[Ibix]

I think you misunderstood me. What I was saying was that the line element is $d\tau^2=g_{00}dt^2+\mathrm{other\ terms}$, and hence that the time dilation factor is $\sqrt{g_{00}}$. That is, $g_{00}$ is the square of the factor you are looking for. You appear to have done the algebra correctly in #14.

 

[Dukon]

OIC

when you say time dilation factor you mean first order in $\tau$ which means we both agree yes $$ \eqalign{
d\tau^2\ &=\ \ g_{00}\ \ dt^2 \\
d\tau\ &=\ \ \sqrt{ g_{00} }\ \ dt \\
\tau\ &=\ \ t \ \ \sqrt{ g_{00} } \\
}
$$ physics aligned; language misaligned. Langauge is a true hindrance! (for me anyway)

I think I might be wrong about the sign of the ${r_Q}^2$ term in my post #14. I think when expanded everything under square root needs to be negative not positive

 

[PeterDonis]

I think I might be wrong about the sign of the ${r_Q}^2$ term in my post #14.

You were in your "Kerr-Newman" equations in that post, yes. The sign of the $r_Q$ term is opposite of the sign of the $r_s$ term.

I think when expanded everything under square root needs to be negative not positive

No, when expanded (as in, for example, your "Reissner-Nordstrom" equation in post #14), your signs were correct; the sign of the $r_s$ term under the square root is negative and the sign of the $r_Q$ term under the square root is positive.

Note also that, for your equations to be valid, the overall quantity under the square root must be positive. Physically this means your equations only work outside the horizon in Schwarzschild or Reissner-Nordstrom spacetime, or outside the static limit (the boundary of the ergosphere) in Kerr or Kerr-Newman spacetime.

 

[Dukon]

Correction to post #14 $$\eqalign{

\tau &= t \sqrt{1 - \frac{r_s}{r} } \hspace{1.7in} \ J=0 \ Q=0 \ \ {\rm Schwarzschild} \cr

\tau &= t \sqrt{1 - \frac{r_s r}{\rho^2} } \hspace{1.6in} \ J\ne 0 \ Q = 0 \ \ {\rm Kerr} \cr

\tau &= t \sqrt{1 - \frac{r_s}{r} + \frac{{r_Q}^2}{r^2} } \hspace{1in} \ J=0 \ Q\ne 0 \ \ {\rm Reissner-Nordstrom} \cr

\tau &= t \sqrt{1 - \left\{ \frac{r_s r - {r_Q}^2}{\rho^2} \right\} } \hspace{1in} \ J\ne 0 \ Q\ne 0 \ \ {\rm Kerr-Newman}.

}.$$