# Are acceleration and gravity completely equivalent?

Hello everyone,

I just mailed this question to NASA's Cosmicopia but I'm afraid they might not be able to reply. So I registered in these forums to see if maybe someone can give some guidance.

So here it is:

We know that accelerating a body with mass to the speed of light requires infinite energy. So far I understand, for all accounts, light would behave in relation to an extremely accelerated body as it would in relation to a very massive body, making both situations analogous. If it is true that making a body have infinite mass requires infinite amounts of energy, how is it that most scientist consider black holes as a possible phenomena?

From my poor understanding of the field, either acceleration and gravity aren't exactly the same thing, or when a super-massive star, in it's last moments implodes, on it's way to it's own core it manages to "somehow collect" the infinite energy required to "accelerate" all it's matter to the speed of light and make it have infinite mass (thus, creating a black hole).

tiny-tim
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welcome to pf!

hello damianpaz! welcome to pf!
… either acceleration and gravity aren't exactly the same thing, or when a super-massive star, in it's last moments implodes, on it's way to it's own core it manages to "somehow collect" the infinite energy required to "accelerate" all it's matter to the speed of light and make it have infinite mass (thus, creating a black hole).

no, a black hole doesn't have infinite mass (or infinite energy) …

every star "warps" space-time slightly (that's why light rays are bent very near a star)

the closer the light rays are to the centre of the star, the more they are bent

obviously, they can't travel through the star, so the amount of bending is limited by the radius of the star …

reduce the radius, and the bending can become greater

reduce the radius beyond a certain limit, and the bending can become so great that nothing can get out

a black hole is a star that is smaller than that limit

(a star that small will actually collapse to zero size, but that doesn't affect the black-hole-ness)

the mass density is very large (arguably infinite), but the mass is the same

Dale
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either acceleration and gravity aren't exactly the same thing
You are correct. They are not exactly the same thing. In particular tidal gravity is different.

reduce the radius, and the bending can become greater

reduce the radius beyond a certain limit, and the bending can become so great that nothing can get out

a black hole is a star that is smaller than that limit

(a star that small will actually collapse to zero size, but that doesn't affect the black-hole-ness)

the mass density is very large (arguably infinite), but the mass is the same
Right, although I think it is simpler to think of the area versus the mass of the star as the radius, unlike the area, is subject to curvature.

So the critical value is:

$$A = 16 \pi G^2 m^2/c^4.$$

E.g. if the area of the (non-rotating) star becomes smaller than this the star will collapse and become a black hole

Hi, thanks for the welcome and the replies.

hello damianpaz! welcome to pf!
no, a black hole doesn't have infinite mass (or infinite energy) …

Oh I forgot about this yes. I understand that the black holes gets formed from a finite amount of mass. What I find hard to understand is how it gets to get as dense as to being able to accelerate light "backwards" or at least cancel out light's velocity. To my mind this implies that gravity can pull harder than the speed of light.

You are correct. They are not exactly the same thing. In particular tidal gravity is different.

Hi! Is there some analogy to explain this to average joe? (I'm average Joe, well kind of :) )

Dale
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The equivalence principle says that over a small region of spacetime gravity and acceleration are equivalent to first order. In other words, over very small distances physics in a freely-falling frame in gravity is the same as physics floating in space far away from gravity. However, over large enough distances you can physically detect the fact that gravity doesn't point the same direction everywhere and isn't the same strength everywhere, which is called tidal gravity.

The equivalence principle says that over a small region of spacetime gravity and acceleration are equivalent to first order. In other words, over very small distances physics in a freely-falling frame in gravity is the same as physics floating in space far away from gravity. However, over large enough distances you can physically detect the fact that gravity doesn't point the same direction everywhere and isn't the same strength everywhere, which is called tidal gravity.

Maybe tidal gravity could be analogous to acceleration caused by centrifugal force. Still, to the effect of making a body with mass infinitely dense by the means of acceleration it'd be required infinite amounts of energy isn't? Does this also applies to acceleration by gravity?

A fast moving object does not turn into a black hole, if that is what you are thinking. See: http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/black_fast.html

Someone suggested that in a thread at gamedev.net yes, it did cross my mind but I'm actually more interested in trying to understand why is it impossible to accelerate an object until it has "black hole" amount of density but black holes themselves with infinite density (?) are considered to be possible. From there my assumption that maybe gravity may have some sort of property that cannot be replicated by adding energy to a body with mass.

I'm not questioning science. I'm just trying to push my brains a little further without becoming a scientist myself :)

Dale
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I'm actually more interested in trying to understand why is it impossible to accelerate an object until it has "black hole" amount of density but black holes themselves with infinite density (?) are considered to be possible.
Because it requires an infinite amount of energy in the first case and a finite amount of energy in the second case.

Because it requires an infinite amount of energy in the first case and a finite amount of energy in the second case.

That's simple enough. Then I guess that's where the analogy ends. Given enough gravity, you can make a finite amount of mass infinitely dense. No other force in the universe, SFWK, can accelerate (compress) mass that way.

Given enough gravity, you can make a finite amount of mass infinitely dense.

Not really so....but perhaps "arguable" as tinytim posts.....there is not insofar as we know anything proven 'infinite' within a black hole.....space and time seem to disappear as independent entities... light cannot escape, but our mathematics at the singularity within a black hole is incomplete....

In fact the contray is true: everything seems FINITE within a black hole...according to the holographic principle:

try reading here if interested: http://en.wikipedia.org/wiki/Holographic_principle

[In simple terms, information (and therefore everything else) within a black hole is finite. A black hole seems to be a FINITE repository.]

Not really so....but perhaps "arguable" as tinytim posts.....there is not insofar as we know anything proven 'infinite' within a black hole
I think the volume between the EH and a sphere passed the EH goes to infinity if the sphere is approaching the singularity. Do you think I am wrong about that?

our mathematics at the singularity within a black hole is incomplete....
I think the theory of GR breaks down, but not the mathematics.

Regarding the above post:

Here is how Lee Smolin describes it:

In nature we have yet to encounter anything measureable (I think he means observable) that has an infinite value. But in both quantum theory and general relativity we encounter predictions of physically sensible quantities becoming infinite.....at the point (inside a black hole) at which density becomes infinite, the equations of general relativity break down.

( THE TROUBLE WITH PHYSICS, pg 5)

Quite possibly my ad hoc description previously posted isn't precisely correct....All I know is that everywhere I look (Susskind, Greene, Kip Thorne,etc) big bang singularities and black hole singularities are generally agreed to confound gr and quantum mechanical calculations....neither works, hence the desire for quantum gravity, "unification"...