# Crackpot Physics Question of the Week

Though the Crackpot in question is myself.

Occasionally, while my mind wanders, I’ll hit upon a notion in some field of science in which, while I have had a few college courses, I am no expert.

My question is this. If you have two black holes drawn towards each other, could they accelerate past the speed of light?

My reasoning : As an object approaches the speed of light, its mass increases… normally this means it requires more energy to accelerate, and provides the universal speed limit of C.

But with acceleration due entirely to gravity, the mass increase of the two black holes would serve to intensify the force applied as well. So as they reach infinite mass, their gravitational pull becomes infinite as well. Not to mention the fact that they’re getting closer all the time, further increasing the pull.

If it is possible that they could exceed C, what are the implications? How massive would each black hole need to be to achieve the effect?

As the gravitational force approaches infinity, speed approaches c. But there’s nothing particularly special about this case that allows it to go past c. In fact we don’t know of any circumstances that allows it.

Is there an equation for acceleration due to gravity in the Einsteinian mold? The one I know for classical mechanics doesn’t have the c-limiter, of course.

it really isn’t a crackpot question at all and a quite good one. To begin with I’m not sure if gravitational force is based on relatavistic mass or only on rest mass. For thesake of argument let’s say gravitational force increases proportional to relatavistic mass which increases with relative velocity. The problem is that gravity is a very weak force and when you apply distance to the inverse square law the force is quite weak until the two objects are extremely close. It’s oversimplifying but force of earth’s gravity is half when you are only one half earth diameter (about 4k miles) away from the surface.

Another important thing to remember when you’re talking about General Relativity is the difference between the “cosmic speed limit” and the motion of space itself. What General Relativity says is that you can’t outrun a light ray that was emitted at the same spacetime location as you. However, this property of what the “local speed of light” is is a property of spacetime itself, and it’s possible for space to expand “faster than the speed of light.” A good example of this is the expansion of the Universe: by Hubble’s Law, the speed at which we see distant galaxies recede is something like v = Hd, where d is the distance to the galaxy and H is called “Hubble’s constant.” It’s easy to see that if d is large enough, these distant galaxies would be moving away from us “faster than the speed of light”. Of course, they’re not really moving faster than light; it’s just that space is expanding “faster than the speed of light” between here & there.

So to answer your last question: while it’s possible that a distant observer might see two black holes moving “faster than light” towards each other, this doesn’t necessarily mean that the cosmic speed limit has been broken.

As to your first question, I don’t have my notes on relativistic dynamics with me, but I suspect that the “force” would grow at the same rate (or slower) than the “mass”, meaning that the acceleration would stay constant (or even decrease.) I’ll jot down some notes later today and see what I can figure out. (Of course, the full case of two black holes colliding, even head-on, is unsolved in General Relativity, so don’t expect too much.) Well, more fundamentally there’s really no such thing as a gravitational force in GR. Forces as we know them from high school physics really don’t exist at the bleeding edges of physics.

In GR, all you “really” have is the relative acceleration of the geodesic worldlines of two particles. As an example, the Sun’s gravitational influence on the Earth:
Set up two spacetime geometries for the area around the sun. One has a sun and one doesn’t. Determine the geodesics in each of a particle starting with a certain position and 3-velocity. Now, if possible (and it’s really not clear that you can do this in more but the simplest situations), determine how points in the two geometries correspond to each other. Measure how fast the two curves separate. This is the acceleration due to gravity, from which an effective force can be worked out.

You’ll need to take into account the gravitational waves generated by the moving black holes.

The same thing happens to orbiting neutron stars. The faster the objects move the more gravitational energy they shed.

Maybe Chronos will drop by and give a more definitive answer.

Bypassing the question as to whether force is a legitimate concept in GR just remember that 1) relativistic mass is a very poor concept, (it causes confusion as shown by this question) and 2) an object is not going to undergo gravitational collapse due to its motion.

If this were not true then a relativistic subatomic particle would cause most stellar objects to collapse to black holes.

The hole in the OP’s argument (pun intended, sorry) is the use of the word “infinite.”

Black holes form from a finite amount of mass. In fact, all the total mass in the observable universe is a finite amount.

All the theories of physics we deal with come with a basic provision that an infinite amount of mass and an infinite amount of force are impossibilities. The very reason that massed particles can never, under any circumstances, even theoretical ones, reach the speed of light is that is would take an infinite amount of force to do so, and the entire universe combined cannot create this force.

So the black holes in the OP cannot reach C. And they most certainly cannot exceed C.

The standard physics of today can handle the case of two approaching black holes fairly easily. It turns out that they merge and form a larger black hole. Nothing mysterious happens.

Is this a crackpot question? Well, let’s put it this way: whenever you use the words “accelerate past the speed of light” in a physics question, you need to examine your assumptions to see where you are going wrong.

Whoa. I never argued anything… I asked a question, and explained why the question was asked. I’m not saying ‘Einstein was wrong! Black Holes can travel at Warp 4! Long Live Captain Picard!’

And while the current structure of relativistic theory is something I accept as true with my partial understanding, it’s not like, gospel, or anything. It may someday be invalidated. So, please, dial down the snootiness just a bit, if you don’t mind.

Hm. Let me clarify my objective with the question : according to the equations governing relativistic motion, what is the maximum velocity a pair of black holes will obtain when attracting one another, assuming optimal distance and initial mass?

Obviously, when two black holes collide, they form one black hole. I’m not dense. (There’s a pun for ya!) I don’t care about the collision though. The collision has nothing to do with what I’m asking, apart from the fact it represents the time-point when they stop moving towards one another.

According to simple Newtonian gravity, Force = GravConstant * Mass1 * Mass2 / DistanceSquared. So as these objects accelerate towards one another, DistanceSquared begins to decrease rapidly … and at higher velocities, both Mass1 and Mass 2 will begin to increase, according to relativity. In classical mechanics, we also have Force = Mass * Acceleration.

So for Mass1 , it’s own Mass cancels out when calculating its Acceleration.

My impression (perhaps flawed) of the Relativistic speed limit was that this increasing mass was what kept objects below C, because as they approach C, their mass begins to increase instead of the velocity.

According to Classical Mechanics, this just cancels out. Obviously, Classical Mechanics has to be wrong, in this case… but how does Relativity handle it? Force of gravity in relativity still involves the mass of the respective objects, does it not?

Just simply saying ‘They can’t exceed C’ really isn’t good enough. I guess I’m looking for the Mathematical upper limit of their velocities in ideal conditions. Perhaps it’s ‘infinitely close to C’, based on the Relativity equations. That’s fine, but I’d like to see why that is.

I’ve thought of an interesting experiment you could use to test your theory: observe the buttocks of Roseanne Barr in motion. Perhaps not infinite mass, but its as good as you’re going to get in a finite universe.

So what you really want is the relativistic mass equation. No problem, except for formatting. m[sub]0[/sub] is the original mass.

m = m[sub]0[/sub]/(1 - v[sup]2[/sup]/c[sup]2[/sup])

See what happens? As the velocity increases, mass also increases.

The maximum velcocity shows when you solve for v.
v = (c[sup]2[/sup] - [m[sub]0[/sub]/m] * c2)[sup]1/2[/sup]

As the relativistic mass, m, gets larger and larger, the second term of the equation gets smaller and smaller.

So v -> c, but never can actually reach it.

And of course I never called you a crackpot - which was your word to begin with. I just said you needed to examine your assumptions.

I think something that’s missing here is that “relativistic” mass is a hazy concept. An object really only has one mass: its rest mass. If you’re sitting next to a boulder, staying in the same position relative to it, you can measure its rest mass m[sub]0[/sub]. Now, if you’re moving past it at some fraction of the speed of light v (in units so that c is 1) and measure its mass, you’ll get a different answer: m[sub]0[/sub]/(1-v[sup]2[/sup]). The boulder hasn’t gained any mass, it just looks heavier when you’re moving past it.

Actually, nowadays the concept of relativistic mass is not longer used because it doesn’t have a real physical meaning. Of course you can define a magnitude m := m[sub]0[/sub]/(1 - v[sup]2[/sup]/c[sup]2[/sup]), but it is not a real mass. From Taylor, Wheeler, Spacetime Physics:

In simpler terms, I think this is a good way to see it. An object has two masses, a gravitational mass, and an interial mass. The gravitational mass is how much the object distorts spacetime. The interial mass is how much energy it takes to accelerate the object. Most of the time, these are the same. But when you go close to c, your inertial mass increases, i.e. it takes more energy to accelerate you, but your gravitational mass doesn’t change. Newton’s equations use gravitational mass, so gravitationally speaking, the masses of the black holes don’t change.

[Beats head against wall.]

Guys, I called it the relativistic mass equation because that’s what it’s commonly called. If the OP wants to do further searching on the subject, that’s probably the best term to use. In fact, googling on it brings up a huge number of good sources of information, even for someone like the OP who doesn’t have any basic understanding of relativity theory. Nitpick away, but ease of understanding comes first.

However, I don’t think it’s a nitpick to say that Yumblie is wrong to make his distinction.

The Fabric of the Cosmos : Space, Time, and the Texture of Reality, by Brian Greene

That equation was exactly what I was wanting. Though now that I see it here, I seem to recall having seen it before… in the back of my mind, maybe, I was thinking of it as a velocity equation rather than a mass equation.

So is there also an equation in Relativity for gravitational attraction between two objects?

And I didn’t say you called me a crackpot, I just objected to the tone of your post.

Also, for the record, I think I more or less understand ‘big picture’ relativity… it’s the details, like the specific equations, that I have had little or no exposure to.

It is a velocity equation.

But it is also a mass equation. It’s a floor wax, and a dessert topping.

The real mass equation is m = constant The mass of a particle does not change, it remains a constant and it is the same in all frames of reference. What happens is that mathematically you can explain the increase in the momentum-energy of a particle with the artificial definition of a mass that gets bigger with velocity, and thus you would keep the old formula for momentum, p = mv where m would be that “relativistic” mass. But it`s not a real mass, in a physical sense.