Ok look. The system is going to have a total energy content of X. That’s the combined gravitation potential energy and kinetic energy of the two objects. It’s not really relevant that the objects are black holes seeing how we can only observe the event horizons surrounding the singularities. Say the entire system energy becomes kinetic. The relatavistic KE equation is KE = moc^2 [1/(1-v^2/c^2)^.5-1] so solve for v you get

v^2 = c^2 [1-(moc^2/(KE + moc^2))^2].

Notice how this winds up being c*(1-u) where u is positive?

I now wait to be savaged, and hopefully told how to do subscripts and superscripts.

You can use subscripts and superscripts with the code [sub] and [sup] The easiest way to learn this kinf of things is by looking at another member´s post with the reply function. (At least that`s how I do it)

I wish I could’ve taken a special or general relativity class (low budgets mean they weren’t offered every year), but I didn’t. So I’m no expert, but I do have a physics degree. I believe, however, that the “increase in mass” as an object reaches higher speeds is kind of a false notion.

An outiside observer sees that it becomes more difficult to accelerate an object already travelling really fast, and since that’s the way we measure the mass, says the mass increases. But my department chair told me once that this is really an effect of ENERGY. It takes more energy to accelerate a fast moving object to a certain faster speed, and the equation has a gamma ( =1/sqrt[1-v^2/c^2] ) multiplying the m0 (rest mass). (For instance, it’s really E= (gamma)m0 c^2.) So in some sense you can think of the mass changing: (new mass) m = (gamma)m0, (like E=mc^2, with the new mass).

Also, it seems to reason that the gravitational force uses the rest mass in it’s equation, and so the force of the 2 black holes will be dependent only on their rest mass and their distance of separation, not on their relative speeds (what’s happened before). HOWEVER, how much of an increase in speed this force creates (over a certain distance) DOES depend on their relative speeds (actually, we should be saying velocities).

So, once again, the idea of mass increasing is sort of bunk. No infinite mass in the gravity equation, sorry. Besides, what’s being a black hole really have to do with it then. If this could happen between 2 black holes, why not 2 atoms in space (sufficiently separated, I suppose).

I chose Black Holes as it’s easier for me to conceptualize them attracting each other over vast distances. Two particles, I would expect, would have to be fairly close to each other relative to their distance to any other matter, to achieve this situation… whereas the gravitational pull of a black hole is going to extend a bit farther.

The only thing that matters is the mass of the bodies in question. Gravity has infinite range. Besides, in standard physics tradition we’d assume that there are no other bodies affecting the system at all.

While this is true, I somehow envision the Black-Hole scenario as ‘more likely to occur in a real universe’ … and while gravity has infinite range, its pull drops off sharply as distance increases… my point being that if particle A and particle be were x distance apart… then there can’t be anything closer than x to either particle that happens to have more mass than they do, or the set-up is ruined.

With Black Holes, it’s likely they’re the biggest, baddest gravitators in the neighborhood.

I repeat: to answer your question a physicist would make a model assuming that the two bodies are the only ones in existance. As for black holes being the biggest around: maybe in the “real universe”, but I really see no arguments other than hand-waving for that.

I realize what a physicist would do, but I was explaining my choice of objects for the particular example, not suggesting that a physicist would try to work in a realistic environment. Heaven forbid.

While my OP has been answered to my satisfaction, more or less, I’d still like to see the relativity-theory equation for gravitational attraction between two objects… and I’m also getting some varied answers about whether or not the mass of the object really, truly increases … some seem to say yes, some seem to say no.

So, is there a physicist in the house? Does the gravitational mass of the object increase as it nears c?

There is no gravitational force. It is a fiction designed to explain the observed motions under gravity in terms of Newtonian physics. This is an inherently relativistic question, so It must be handled with GR Here, I’ll roll up my sleeves and try giving an overview of how GR solves the problem (at least how this mathematician understands it; maybe Chronos will correct me).

GR posits that “gravity” is an epiphenomenon of geometry. That is: the universe is a locally-Minkowski 4-manifold within which bodies travel along timelike geodesics. Making a GR model is a geometry problem. We’ll assume there are only the two bodies in question, and that the universe is asymptotically flat.

First, we must determine the initial 3-geometry: the starting state of a 3-dimensional spacelike slice of our 4-geometry we’re modelling. To simplify the question, we will assume that the bodies are nonrotating black holes. This is only because black holes are amazingly simple objects to work with. The geometry outside the event horizon of a nonrotating black hole is completely determined by its mass, and the geometry inside doesn’t matter (HA!). The outside geometry in a slice through the center of the hole looks like the surface z=K sqrt(r-C) (note the hole of radius C). The radius of the hole is completely determined by the mass.

Now, given two of these holes, we must put them together in a single universe. We can slice a plane through the center of both holes and demand that the system be rotationally symmetric about the axis containing both centers. This slice will thus completely determine the initial 3-geometry. Again to simplify we’ll assume the holes are of equal mass so the initial condition can be symmetric about the center of mass. We need to deform each surface so we can sew them together with a smooth seam. This can be done, but it’s hairy.

Now we’ve got a deformed plane with two holes of radius C centered at (x[sub]0[/sub],0) and (-x[sub]0[/sub],0) (without loss of generality) in the “naive” coordinate system looking at this like a graph of a surface. Really we have a 3-manifold with a rule for determining the length of any tangent vector to the surface, but this graph gives a good heuristic.

Now what must be done is to plug this manifold in as the initial conditions of the Einstein equation. The Newtonian approximation will not work as it is a simplified form of the linearized Einstein equations, which only apply to weak gravitational fields. The forward solution of this equation will give a 4-manifold describing the future geometry of spacetime. The two parameters we have are the mass of the holes (both the same, remember), and the x[sub]0[/sub] measuring initial separation. By the nature of the equations (waves hands) no future-oriented geodesic starting on the initial surface will even become timelike, let alone spacelike. Thus no part of the holes could “move at the speed of light”.

The gravitation is not covered by special relativity, you need the General Theory of Relativity. The equations of the relativistic theory of gravitation are quite complex. They require tensor notation and non-Euclidean geometry.

But we an outline some of its principles: as John Archibald Wheeler said: “Matter tells space how to bendm, and space returns the compliment by telling matter how to move”
Five principles:

1-Mach’s Principle

2-Principle of equivalence (gravitational mass = inertial mass)
→ No local experiment can distinguish nonrotationg free fall in a gravitational field from uniform motion in the absence of any gravitational fields.
->Local experiments cannot distinguish between being at rest in a uniform gravitational field and undergoing uniform acceleration in the absence of any gravitational field (i.e. a rocket, a space elevator…)

3-Principle of covariance-> All observers, inertial or not, see the same laws of physics (this calls for tensors, because they are geometric objects defined independent of any coordinate system)

4-Correspondence principle->in weak gravitational fields, we recover the special theory.

5-Principle of minimal gravitational coupling

With this, we get an equation that basically says:

Well, thanks Desfibrador and Mathochist, now I have to actually go learn tensor notation and non-Euclidean geometry. Sigh. Oh, well, sacrifices must be made.

Reminds me of the line from Three Amigos.

Lucky : “Well, we’re just gonna have to use our brains.”

Well, actually some things can be understood without all the mathematical artillery. There are some good books that offer a qualitative view of the theory… I think I read some years ago a little book by Einstein about the Theory of Relativity (both the special and the general) that could be easily followed with just High School physics.

However, if you want to learn more you should start by the Special theory and a great book about it, very “user friendly” is Taylor, and J.A. Wheeler Spacetime physics I recommend this book to anyone interested, you don’t need neither advanced Physics nor Mathematics to follow it. Unlike many Physics books, it’s not filled with equations, but is mainly text, lots of examples…

Well, be careful. A lot of what’s called “tensor notation” is heavy index juggling that doesn’t make clear that the objects are independant of coordinates. Also, most references to “non-Euclidean geometry” will get you hyperbolic geometry. What you want is “Riemannian geometry”, “pseudo-Riemannian geometry”, and “differential geometry”.

An excellent book, imho, for all things GR-related is The Phone Book (a.k.a.: Gravitation). It looks like the monolith from 2001, but it’s actually very approachable. It has clearly marked “tracks” to make a first pass and a more in-depth study, plenty of material on stellar structure, black holes, and cosmology, and more examples and exercises than you can shake a vector at. It also really emphasizes the geometry, and one part is basically an introduction to the differential geometry needed (one chapter giving a working knowledge in Track 1, something like eight giving a very decent buildup of the theory in Track 2).

Hey “O Desfibrador,” do you know if that Feynman quote is for real? A certain physics department I know used it for a T-shirt a few years ago, but the word I heard was that he said something similar, but perhaps not the same. Actually, I think the exact quote on the shirt was “Physics is like sex, sure there are some practical applications, but that’s not why we do it.”