Two black holes separated by more than one Schwarzschild radii

[kmarinas86]

If you have two black holes separated by more than one Schwarzschild radii, given a location at:

1. The event horizon of one of the black holes.
2. A location between both black holes
3. A location on the axis that passes the centers of both black holes.

Wouldn't the potential there actually be less than it would be if the objects were isolated?

First, note the black holes are pulling in opposite directions, so this isn't simply the sum of the gravitational potentials (is it?), but rather a sum of their vectors! If two black holes were about to collide, couldn't the matter that is between them race out between black holes A and B? Surely the velocity the particle will encounter will only increase. With respect to our frame of reference (and time perspective), is it possible that the matter (which appears to be forever collapsing to the Schwarzschild radius) appear to us as being sucked out as the black holes approach each other?

Could the jet coming out of a black "hole" actually be the stream of particles exiting the region near the axis of rotation of two black holes? Remember that from our time perspective (our time coordinate), the matter falling in never seems to cross the Schwarzschild radius, so in a finite amount of time, the matter may be expected to be jutted out in jets (because the black holes would be orbiting each other and the vectors between them cancel). If the masses were unequal, would this stream (between the unequal masses) actually be jutted outside the axis of rotation, causing the jet to be a helix in shape? If the particles were charged a certain way, then helix could expand and become very wide. Because of their relativistic velocity, the expansion would be mostly horizontal rather than vertical.

Perhaps I am wrong when I say the vectors cancel each other. If they don't cancel, then what do they do?

 

[krateesh]

separated by < one swarzchild radii ? Are they identical ?
i really culdn't understand that question statement only...

 

[kmarinas86]

http://www.geocities.com/kmar86/gravitationalfield.xls

I made an Excel file that I think represents what I'm talking about. Here is the math behind it:

$F_x^2+F_y^2=F_r^2$

$\left(F_r cos\left(\theta\right)\right)^2+\left(F_r*sin\left(\theta\right)\right)^2=F_r^2$

$F_r\propto\frac{1}{r^2}=\frac{1}{x^2+y^2}$

$cos\left(\theta\right)=\frac{x}{r}=\frac{x}{\sqrt{x^2+y^2}}$

$sin\left(\theta\right)=\frac{y}{r}=\frac{y}{\sqrt{x^2+y^2}}$

$F_x\propto}\frac{x}{\left(x^2+y^2\right)^{\frac{3}{2}}}$

$F_y\propto}\frac{y}{\left(x^2+y^2\right)^{\frac{3}{2}}}$

$F_x\propto\frac{1}{1-\frac{r_s}{r}}$

$F_y\propto\frac{1}{1-\frac{r_s}{r}}$

 

[Chris Hillman]

Are you sure you mean "potential" here?

Hi, kmarinas86,

If you have two black holes separated by more than one Schwarzschild radii, given a location at:

1. The event horizon of one of the black holes.
2. A location between both black holes
3. A location on the axis that passes the centers of both black holes.

Wouldn't the potential there actually be less than it would be if the objects were isolated?

You'd have to explain what you mean by "potential" in the context of gtr, but in Newtonian gravitation, if you have an isolated system composed of two roughly spherical objects, then indeed, if you plot the equipotential surfaces in "space" in a comoving coordinate system, these look like nested Casini ovals, i.e. in cross section, a sepatrix shaped like a figure eight separates two separate sets of nested equipotentials shrinking onto the two objects, from exterior equipotentials which rapidly become spherelike as you move away from our isolated system.

See also http://www.math.ucr.edu/home/baez/RelWWW/weylvac.html for a discussion of a kind of potential which can be defined (with suitable caveats) for static axisymmetric vacuum solutions in gtr (the Weyl vacuums). As it happens, in a Newtonian context, the picture of "the Potential for Two Positive Mass Monopoles on a Massless Spring" happens to shows the above mentioned exterior equipotentials becoming approximately spherical as you move away from the isolated system.