# Black holes, speed of light and gravity oh my!

lets say we had a black hole or something that created a large gravitational field , one that could accelerate something to near the speed of light , so we would have a certain amount of mass or energy that could accelerate our object to near the speed of light as close as we could get it , if we could pack in more mass into this singularity of the black hole so then would this added mass or energy have no effect on accelerating the object because it would be reaching its terminal velocity . or would this be like the amount of energy in the universe .

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russ_watters
Mentor
That's one long and incoherent sentence but maybe this helps: an object can never reach the speed of light but as long as you keep applying a force it will keep accelerating. There is no terminal velocity like you are describing.

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how can you keep accelerating something and yet it will never reach the speed of light ,
and how could it not reach a terminal velocity.

As Russ said, a black hole doesn't really have a terminal velocity. The terminal velocity we see on earth is because of air resistence.

our terminal velocity is the speed of light

DaveC426913
Gold Member
OK, let's clear a few things up:

1] Acceleration due to gravity and acceleration due to propulsion are synonymous. Whether you accelerate by using your engines or you accelerate by falling in to a black hole has no effect on how close to the speed of light you can get. All I'm trying to say here is: the BH does not add a complication to accelerating to c.

2] Velocities do not add linearly at relativistic speeds. If you are travelling at .9c and you (either turn on your engines or fall into a BH), you will not "hit" 1.0c. What happens is your velocity goes from .9c to .99c. If you continue to accelerate, your velocity will reach .999c. You can accelerate until you grow old, and your velocity will reach .999999999c.

The formula is $$v = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$

So, c is not a terminal; it is an asymptote.

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LURCH
how can you keep accelerating something and yet it will never reach the speed of light ,
and how could it not reach a terminal velocity.
Suppose you are travelling atr very nearly the speed of light; only 2kph slower. You can apply an acceleration force, and cover 1/2 the difference. Now, you are just 1kph slower than c. Apply more acceleration, and oyu make up half the difference again; you are now travelling c-.5kph. You can keep accelerating through c-1/4, 1/8,etc...

you continue to accelerate, but never reach c. And so, no lightspeed and no terminal velocity.

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Or, y'know... what Dave said.

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If you go from a velocity of β=0.999 to β=0.9995, the γ goes from about 22 to 32; i,e, the energy increases by ~45%. It is very nonlinear in the amount of energy required.
Bob S

ok i think i understand, so if i kept adding mass to the black hole i would just keep getting closer to the speed of light .

Suppose you are travelling atr very nearly the speed of light; only 2kph slower. You can apply an acceleration force, and cover 1/2 the difference. Now, you are just 1kph slower than c. Apply more acceleration, and oyu make up half the difference again; you are now travelling c[/ki]-.5kph. You can keep accelerating through c-1/4, 1/8,etc...

you continue to accelerate, but never reach c. And so, no lightspeed and no terminal velocity.

Forgive me if I'm misunderstanding, but I'm not sure this is quite right. The above is a statement of Zeno's paradox. It's the classic paradox involving a man standing near a wall. To reach the wall he must first cross half the distance, and then half that distance, ad infinitum. Thus he can never reach the wall. The fallacy can only be exposed by the mathematical notion of functional continuity; motion is continuous, and shouldn't be discretized as the paradox describes.

Here we're dealing with velocities, but the idea is the same. Velocity is continuous, so discretized increases in speed isn't the reason we can't reach light speed. The reason is because the relativistic energy is proportional to the gamma factor, and as speed increases, the gamma factor goes to infinity. Thus it would require infinite energy to accelerate any massive object to c.

Again, sorry if I'm misunderstanding what you said.

ok i think i understand, so if i kept adding mass to the black hole i would just keep getting closer to the speed of light .
"You" would never get anywhere in the vicinity of c, though what is left of you after gravity has ripped you to pieces will be accelerated to greater speeds for greater mass black holes. Exactly what speed you reach at a certain radial distance from the BH depends on where you started from.

Nabeshin
So, c is not a terminal; it is an asymptote.
To be fair, a terminal velocity as created by air resistance is also an asymptote.

DaveC426913
Gold Member
To be fair, a terminal velocity as created by air resistance is also an asymptote.
Yes, I didn't mean to imply it wasn't. I was trying to disabuse the OP of the notion that, wrt reaching c, acceleration increased until some point, as then just stopped.

DaveC426913
Gold Member
...what is left of you after gravity has ripped you to pieces...
With a large enough black hole, the gradient is small enough that gravity will not pull you apart.

Nabeshin