Acceleration of Particles: Black Holes & Speed of Light

[Allen_Wolf]

I have a little doubt to get cleared.
Gravity of the Earth causes an acceleration of 9.8 m/s2 which increases the velocity of a body at free fall.
So for a very massive body like a black hole FOR INSTANCE, the acceleration due to gravity on a body will be very high, right? So if a body falls into it from a specific distance with an initial velocity u, couldn't it accelerate to the speed of light? or a velocity 99.9% of the speed of light?
Please forgive me if it doesn't make any sense

 

[PeterDonis]

Gravity of the Earth causes an acceleration of 9.8 m/s2 which increases the velocity of a body at free fall.

Relative to the Earth, yes.

So for a very massive body like a black hole FOR INSTANCE, the acceleration due to gravity on a body will be very high, right?

Only if you get close enough to it. If you have a black hole with the mass of the Earth, and you are at a radial distance from it equal to the radius of the Earth, the acceleration due to gravity on you will be 9.8 m/s^2, just as it is on the surface of the Earth. But because the hole itself has a much smaller radius than the Earth, you can get much closer to it and still be outside it, which will produce a larger acceleration due to gravity.

couldn't it accelerate to the speed of light?

No. More precisely, if you consider a series of static observers (observers "hovering" at a constant altitude above the black hole), and have each of them measure your speed relative to them as you fall past them, each of them will measure a speed less than the speed of light (but closer and closer to it as the observers get closer and closer to the hole's horizon).

or a velocity 99.9% of the speed of light?

Yes, there will be some static observer, at some distance above the horizon, who will measure you to have such a speed as you fall past him.

 

[Allen_Wolf]

So what happens when a body accelerates to 99.9% the speed of light?
Will some of the matter get converted to energy or something?

 

[PeterDonis]

So what happens when a body accelerates to 99.9% the speed of light?

Nothing. Remember we are talking about relative velocity; velocity is not absolute. The body itself feels no force and is in free fall, so it does not undergo any change.

Will some of the matter get converted to energy or something?

No. See above.

 

[jbriggs444]

So what happens when a body accelerates to 99.9% the speed of light?
Will some of the matter get converted to energy or something?

As measured from a "stationary" frame of reference, the body will be measured to be shortened in the direction of travel. Its clocks will be measured to slow down (that's time dilation). Its kinetic energy will be huge, larger than the energy equivalent to its rest mass.

But all of these things are artifacts of our choice of reference frame. There is no intrinsic effect on the object itself. As far as the object is concerned, its dimensions are unchanged, it clocks are unchanged, its mass is unchanged and its energy is unchanged. Right now as you read this, you are moving at 99.9% of the speed of light relative to something that is moving at 99.9% of the speed of light relative to you. Do you feel any different?

 

[Chris Miller]

As you approached c, the distance to the black hole would foreshorten. To you it might appear miles away, while to an Earth observer, possibly light years. I have no clue how its gravity would be impacted by your foreshortened distance.

 

[jbriggs444]

As you approached c, the distance to the black hole would foreshorten. To you it might appear miles away, while to an Earth observer, possibly light years. I have no clue how its gravity would be impacted by your foreshortened distance.

If you are contemplating a black hole, you are assuming general relativity. In general relativity, gravity is part and parcel of space-time curvature. It is independent of coordinates. If you are falling freely into a black hole, there is no local experience of gravity. You remain weightless as the horizon passes you.

 

[Chris Miller]

If you are falling freely into a black hole, there is no local experience of gravity.

Your acceleration would increase with proximity, no?

 

[PeterDonis]

Your acceleration would increase with proximity, no?

Your coordinate acceleration relative to static observers would. But you would feel no acceleration; you are in free fall.

 

[jbriggs444]

Your acceleration would increase with proximity, no?

Be careful. You appear to be applying Newtonian intuitions in an environment where they will lead you astray.

As you freely fall, your proper acceleration is zero. That's true by definition. In another frame of reference using another set of coordinates your acceleration will depend on the coordinates you choose. I may be mistaken, but in Schwarzschild coordinates, your acceleration will reduce toward zero as you approach the horizon. @PeterDonis seems to have a different measure in mind -- your acceleration compared to the nearby hovering observers that you are passing.

 

[PeterDonis]

I may be mistaken, but in Schwarzschild coordinates, your acceleration will reduce toward zero as you approach the horizon.

Your coordinate acceleration, yes. But that's because the Schwarzschild time coordinate gets highly distorted as you approach the horizon.

@PeterDonis seems to have a different measure in mind -- your acceleration compared to the nearby hovering observers that you are passing.

Yes, I was referring to the coordinate acceleration in the rest frame of the static observer; this is different from the coordinate acceleration in Schwarzschild coordinates because of the Schwarzschild time coordinate distortion I referred to above--or, to put it another way, because the static observers close to the horizon are highly time dilated relative to observers at rest at infinity.

 

[Chalnoth]

If you are contemplating a black hole, you are assuming general relativity. In general relativity, gravity is part and parcel of space-time curvature. It is independent of coordinates. If you are falling freely into a black hole, there is no local experience of gravity. You remain weightless as the horizon passes you.

Sort of. This is true for a point-particle. We are not point-particles.

An object with some size to it (e.g., a person) will feel significant effects due to the tidal forces of the black hole, particularly for smaller black holes. Imagine a person falling feet-first near a black hole. They would be stretched as their feet felt the gravitational pull of the black hole more strongly as their head. Due to the same effect, they would also be squeezed from the sides. And if they're falling into a rotating black hole away from that black hole's equator, they'll also be twisted due to the twisting of space-time caused by the black hole.