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At What Speed do Gravitational Waves Propagate?

Updated: May 1, 2021

If gravitational waves do exist, they will propagate at a speed much smaller than the speed of light. Specifically, the speed will be


c=e(2pi/T)^2a^3/h,


where e is the eccentricity of the binary black hole elliptical orbit, T, the Keplerian period, a the semi-major axis and h, the specific angular momentum.


The result can easily be derived from the Schwarzschild metric. Consider the metric


ds^2=dr^2 + G^2d\theta^2.


On this manifold, we consider a flow X=v(d/d\theta_), where v is


v={2E-2M/c^2G}^{1/2}


where E is the constant total energy, and M is the gravitational parameter and c is the speed. In Schwarzschild's case it is conventional to set 2E=1. That the flow be conservative, we must set

v=G'

so that (G')^2+2M/c^2G=2E, where the prime stands for the derivative with respect to the radial coordinate, r. This is easily checked in the case of constant curvature where 2M/G->\rho G^2, where \rho is the constant density in units where the Newtonian gravitational constant is 1. Then G~sin(kr), G'~kcos(kr), and the above expression is easily seen to be the conservation of energy where k^2~1/\rho c^2.


Since G is the determinant of the metric, the Laplace-Beltrami eigenvalue equation is

(1/G)(Gv')'=(G'v'/Gv+v"/v)v+n^2 v=0,

where the eigenvalue n=(M/c^2G^3)^(1/2)=\rho^(1/2)/c, upon transforming to a hyperbolic plane of constant curvature, -G"/G->k^2. That is, M/G^3 becomes a constant density and equal to the inverse square of the Newtonian freefall time.


Consider a binary black hole (BBH) elliptical orbit in velocity space. The eccentricity is the ratio of the distance from the origin in velocity space, c>0, and the radius R_v=M/h, both of which are velocities. The latter can be written as R_v=(\rho/h)V, where V is the volume of the event horizon. Thus, the speed is

c=e R_v=e \rho a^3/h=e f^2a^3/h,

where a is the semi-major axis of the ellipse, and e is its eccentricity.


The maximum frequency f_m~(\rho)^(1/2), the inverse of the freefall time, which is about 1 kHz. Merging events for supermassive black holes are expected to have much lower frequencies. The governing factor in the expression is the eccentricity, which is usually quite small about 5X10^(-2). The speed is not a constant 3X10^8 m/sec, but, rather depends on the geometry of the elliptical orbit.


In addition, we can say something about the orbits to expect in the case of the Schwarzschild flow,

v=G'=(2E-2m/G)^(1/2),

where the total energy is E>0. In contrast to the Kepler case, E can be positive and have elliptic orbits. Instead of the energy being 2E=c^2-R_v^2, we have

2E=c^2+3R_v^2.

The only difference between our expression and Schwarzschild is that the latter sets E=1/2.


Now, introducing the law of cosines for the velocity vectors

v^2=c^2 + R_v^2-2R_vc cos\theta,

into the expression for the flow,

v^2=2E - 2m/G,

yields

R_v^2(1+(c/R_v)cos\theta)=m/G.

This is precisely the equation of a Keplerian ellipse since c/R_v=e, the eccentricity and

R_v^2/m, and m/R_v is the semi-latus rectum.


From the above relation for the flow, it would appear that gravity acts negatively. In analogy

with Levi-Civita's fictitious time, we can introduce a fictitious space variable s by

ds=-dr/G.

This has the effect

dv/dr=-(1/G)dv/ds=m/G^2,

and using the energy conservation equation results in

dv/ds=m/G=(1/2)(2E-v^2).

Squaring both sides gives

ds^2=(1/E^2) dv^2/(1+Kv^2)^2,

which, except for a scale change, is just the form given by Riemann, in his inaugural dissertation, for a metric of constant curvature. The radius of curvature,

K^(-1/2)=(-2E)^(1/2),

is clearly imaginary, indicating that we are dealing with the hyperbolic plane of constant curvature. This is Lobachevskian geometry.


If we reduce our flow to one of constant curvature,

v^2=2E-2\rho G^2,

A solution is G=sin(kr), where k=(2\rho)^(1/2), the inverse free-fall time and 2E=k^2>0.

Now integrating the expression for fictitious space,

-ds=dr/sin(kr),

gives

tan(kr/2)=e^(-ks).

This is the celebrated formula that was developed by Bolyai and Lobachevsky. Given a point not on a line, drop a perpendicular to the line from that point. Let the fictitious length s represent the distance to the line, and kr=\Pi be the least angle such that the line drawn through the point does not intersect with the given line to which the segment s is perpendicular to. The two segments are asymptotically parallel in contradiction to Euclid's fifth postulate.


It is interesting to observe that in the Keplerian case, the velocity space which possesses a unique Riemann metric, with constant negative curvature, -2E, requires open hyperbolic orbits and not closed elliptic ones. In that case, we can define a distance

d=tanh^(-1) (1/e)=tanh^(-1) (R_v/c)=(1/2) ln[(c+R_v)/(c-R_v)]

where the inverse eccentricity plays the role of a relative velocity for e>1. As the eccentricity goes from infinity to 1, the distance increases from 0 to 1. It is the velocity c that is to be associated with the propagation of the putative gravitational waves. As we have shown, it has nothing whatsoever to do with the speed of propagation of electromagnetic waves.


According to the above expression for the cross-ratio, the velocity c is the referral velocity just like the speed of light is in the longitudinal Doppler effect. The total energy, on the other hand,

2E=c^2-R_v^2,


is given by the transverse Doppler effect






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