I have expressed my belief based on physics and symmetry that interloping mass can enter and subsequently leave the combined event horizons along the Z-axis of a Equal Mass binary pair of black holes. That conjecture has not changed. Also, computation of combined event horizons presented herein are believed to be correct.
What I have come to realize concerns the temporal effects of gravity propagation delay on the orbits of binary black holes. This effect, discussed in LINK "Gravity_Arrival_Direction.html", is based on Einsteins belief that gravity propagates at the speed of light. This has been a bothersome issue [to me] since I was only sure the orbital longivity of a pair of black holes would exist in perpetuity if gravity acts instantaneously. For the equal mass case the gravity force vector is always perpendicular to the tangential orbital speed which is always constant.
For the unequal mass binary case the gravity force still acts CG-to-CG through the barycenter but the force provides acceleration of each body on approach and deceleration past perigee. At perigee, the mutual gravity force is perpendicular to both bodies. This situation is altered when gravity is considered to travel at finite speed.
For simplicity 'this' discussion will entertain equal mass BH's with circular orbits (e=0). The only force that can act on the masses without changing momentum is a force (applied at all times) exactly perpendicular to the direction of motion of each.
But if you look at the "Effect Of Gravity Propagation" diagram above, clearly, the arriving green gravity wave exerts the force off angle from the Barycenter because propagation time permits the BH to move around the orbit. And the closer the BH's orbit to the Barycenter, the faster the tangential orbital velocity becomes. For two 10SM BH's with CG's located at 10 Rs from Barycenter, the tangential velocity is 0.118c. At CG orbital radius of 1 Rs (tangent individual BH EH's) the tangential orbital velocity is 0.3536c. The closer they orbit, the faster and further they travel around the orbital track before the partner-launched-gravity-wave arrives to exert its influence; and further around the orbital track means greater mis-alignment of the gravity force from the perpendicular to the direction- of-motion.
At first blush it seems that the Nadir component of the total force vector drops off by COS(phi) while the tangential component increases; this would seem to increase the tangential orbital speed while decreasing the Nadir holding radial force by Fn*COS(phi) and let the BH mass climb out away from the Barycenter. On further inspection it is seen that the c*time distance (above) is shorter than the diameter of the Kepler orbit; inverse square law begins to dramatically increase the total vector force and the radial 'holding' component.
I have computed the arrival angle effect (tabulation above).
Inverse square law force vector increase more than
makes up for the COS(phi) loss. Notice the last column (bold) which shows the radial force component divided by
the required Kepler radial component. A ratio of unity would cause the Kepler orbit to be maintained
(-only true if tangential speed is unchanged). For BH CG locations set at 2 Rs from Barycenter the (radial) Nadir
[de-orbiting] force is increased by 1.099 times the Newtonian force. At CG at 1 Rs the Nadir force increases to 1.211
times Newtonian. At 0.1 Rs the mutual radial attraction is 11.9 times the force required to maintain circular Kepler
orbit. Seems clear that closely orbiting small BH's will de-orbit and merge in some fashion. This may be where the
Kerr Disk shaped singularity comes from. The limiting Rs orbital distance seems to be just under 0.0510 Rs where the
BH orbits almost 180 degrees before gravity arrives meeting the opposite BH nearly head-on. This point is interesting
since the major force vector is almost all tangential (speeding up the BH), and very little radial; but the huge inverse
square law force at that proximity still has a radial de-orbit component that is much greater than the required Kepler
orbit attraction buy some 5,000,000,000:1. Obviously heading for infinity at exactly 180 degrees. While the
above strikeout comments are essentially true (except for neglect of enormously increased tangential speed), I have 'since'
found that De-Orbit does not occur because of tangential increase in speed; centrifugal force wins out over the radial
gravity component which causes the OPPOSITE of De-Orbit to occur.
I am thinking to install math in the inverse square law program (The Math Is Now Installed)
to account for the delayed gravity force arrival angle and force magnitude to track mutual de-orbit in
real time. If De-Orbit happens very quickly, then make an animation to portray. If De-Orbit of close BBH's does happen
quickly, then astronomers are not likely to find Binary Black Hole 'Jets' from the cause I have proposed, at least for
small 10SM Binaries. Small BH Binaries are of less interest anyways because of extreme gradient. I will investigate for
larger SM Binaries which will have much longer de-orbit times, which times are yet to be calculated.
The orbital period of (e=0) EQ mass binaries is proportional to the mass for normalized separation expressed in terms
of (R/Rs), Schwarzschild radius multiples. So a very large EQ mass binary would de-orbit more slowly than a small binary
by an amount proportional to the mass ratio. However, the De-orbit force ratio at equivalent (R/Rs) would be identical
for all (R/Rs)
. This would imply that even very large EQ mass Binary BH's may de-orbit in relative short time if
their spacing is anywhere close to a merged event horizon which just begins at CG-to-CG distance of 4Rs. This being
true if my posit on Gravity Arrival Time is correct. I enhanced the 3-body 2D solver code to apply Delayed
Gravity to the binary pair. The code is generalized to handle circular or elliptical orbiting pairs. See Delayed
Gravity Animation Below.
UN-EQUAL mass Binaries would not have anywhere near as severe a de-orbit rate (again, if my Gravity Arrival conjecture is correct) since with M0 very much greater than M1, the M0 gravity wave arrival at M1 will be very close to orthogonal to M1's direction of motion at all times (see LINK-7: Light Speed Effect On Gravity Arrival Direction) explanation chart). So it would seem that greatly UN-EQUAL mass binaries would be much longer lived than EQ mass Binaries.
I must say "Bummer" that it appears the EQ mass Binaries may be short lived (on the way to creating a Kerr BH)
. The EQ mass Binary is more interesting than UN-EQ Binaries due to the Z-Axis force cancellation symmetry creating
a radially (in X and Y) un-perturbed trajectory in and out of the binary along the Z-axis.
The above analytical result (animated) was set up to commence with a nearly circular orbit (e=0.0269). A FILO buffer was created in the code to store the Time and BH CG positions in X&Y for each iteration of the numerical solver. The buffer array size can be varied; was set at D01(0:5000,1:5) with (dt) time steps of 1E-7 seconds. So buffer depth was 0.5 mSec. Code 'start' uses Kepler inverse square orbital mechanics while the 5000 point 5-column buffer is loaded with (Time) and (X0,Y0) plus (X1,Y1) positions for M0 & M1. When the buffer is full each subsequent Time step loads into position 5000 just after shifting the previous 4999 steps down by one. A buffer search following each subsequent time step permits finding the delay time and position where the gravity propagation distance (delay_time*c) arrives at the opposing mass (for both masses). The gravity arrival line(s) are drawn red for M0 and blue for M1. These lines extend [FROM] the CG position of M0 when it's wave was launched [TO] the arrival at M1's CG and vise versa. The gravity line(s) represent the line(s) of force acting between M0 & M1 where they were when the wave was launched and M1 & M0 CG's (notice order) when the wave(s) arrive. The 1st four pics in the animation (no RED or BLUE) are Kepler orbit positions while the D01(*) buffer is being preloaded. For further explanation to relate this discussion to the enhanced EQUAL MASS Binary Black Hole click Cartoon above. The RED & BLUE heavy dashed lines in the cartoon represent the computed RED and BLUE in (by) the analytical animation solver code.
The result(should not have been, but) was a surprise. While the for-shortened gravity travel distance increased the radial component of the force, the tangential component grows very much faster. Both M0 and M1 gain in tangential orbital velocity which causes the opposite of De-Orbit. This result may mean my interpretation of gravity arrival effect (as I have described) is incorrect or perhaps an incomplete picture of the total orbital dynamics. I believe the math algorithm is correct. On the bright side, this leaves open the spectre of long lived EQ mass binary black holes (that don't de-orbit).
And, Of course, none of this (my) analytical work yet accounts for relativistic mass increase, distance contraction,
and time dilation. Hopefully, one step at a time eventually gets one to a measure of understanding; I'll re-read my BH
strike rather than delete any blunderstuff. Still need to put numbers to EQ mass
Binary de-orbit time.
ONE: Gravity propagation delay effect described above assumes that delay works as proposed. And TWO: That Einstein was correct that gravity propagates at 'c'. I think it safe to say no one knows the speed of gravity propagation if, indeed, it is not instantaneous; or perhaps propagation is at a speed greater than 'c'. Given that ONE: is true, de-orbit rate would be reduced for the case of gravity propagation exceeding 'c'. And instantaneous gravity would just provide Kepler orbits, not to mention gravity frame dragging.
This 1998 Article by Tom Van Flandern articulates several concerns of 'c'-Speed Gravity I have voiced here including: Violation of Conservation of Momentum which happened when I included 'c'-Speed propagation of gravity in my 3-Body Solver, 4th paragraph of his Introduction. And "How does Gravity escape a black hole" is discussed in the 6th paragraph of his Introduction. Also see his paragraph titled "Space-Time Curvature And Retarded Potentials", discusses failure of 'rubber sheet model' to mutually attract two static masses. Author puts gravity propagation speed at 2E+10 c. This article surely removes my problems caused by 'c'-Speed propagation of gravity.
What happens if de-orbit R/Rs goes below 0.051 allows opposing masses to arrive and pass the point where the 'other mass' gravity wave was to be launched; this would seem to slam on the brakes since that "johnny come lately" gravity wave would approach the mass from the opposite side therefore work to slow down the mass. Maybe generating a quasar? Hey, if others can posit off the wall, so can I, check out string theory, pizza slice worm hole time travel, branes and multiverses.
I think the Science Channel may be edging in some of my BH thunder. They are now admitting possibility of flight into the event horizon of a black hole. They suggest using a spinning (Kerr) black hole which should have an annular ring or disk shaped singularity. They suggest entry along the spin axis, although ingress along the spin axis was not explicitly mentioned. In my opinion they still may have a couple things wrong; SCI-CH stated they need negative matter to hold the unstable singularity 'worm-hole' ring open; I would maintain centrifugal force (cause of ring in first place) by conservation of angular momentum would hold the ring open. And SCI-CH says they avoid the spaghettification by choosing a very large black hole; while it is true and I have stated it: gravity gradient is mild at the event horizon of a very large black hole, and severe at the EH of a small BH. But spaghettification would still occur when threading the center of a ~0.051 Rs 'ring' for a large BH. Neither did they delve into extreme centrifugal effects of gravity frame dragging when threading the eye of a Kerr black hole. A point I have not heard SCI-Ch mention is mention of possibility that one can enter the EH on an elliptical trajectory staying adequately far away from a point (non-rotating) or Kerr (rotating) singularity and re-emerge from the event horizon. SEE: Proposed Theory, The Last LINK; Successful Escape. I still believe Kepler-like orbits work inside and outside the event horizon.
From AWSTATS, I think Tesla people want more results there instead of BH physicsstuff. Got to get it all.