As a proton falls into a black hole, it accelerates. How large will its mass become as it approaches light speed?

To answer the title question:
The proton will approach light speed as it crosses the event horizon, since that marks the place where light speed is the escape velocity. It won’t quite reach light speed because it has only fallen a finite distance, but it can get arbitrarily close by simply dropping it from a farther distance. As an object reaches light speed, its mass approaches infinity.

As to what the observers might see:
There are two types of time dilation involved. The first type is that two objects (in this case people with clocks) moving at great speed relative to each other will both observe the other person’s clock tick slower than his own. Person A will think that clock A is faster than clock B, but person B will think that clock B is faster than clock A. As the observers approach c, this effect approaches to each person seeing the other clock stand still in time.

The second type is that an object experiencing an accelerating force (such as being close to a black hole) will appear to have a clock that runs slower than an object that is experiencing a smaller accelerating force. An important fact is that both observers agree on which clock is slower.

Since there is a whole bunch of time dilation, I’m not quite sure what each person would see. From person B’s point of view (the guy falling into the black hole), the two types of time dilation appear to counteract each other. Since the event horizon marks where the first type approaches infinity (due to approaching the speed of light) but not where there is infinite gravity (that would be the singularity), I think that the first type would be the dominant factor. Person B would see the entire outside universe stand still in time. This might mean that from his own point of view he will never reach the singularity of the black hole.

I’m even less sure as to what person A will see.

As I am reading more about it, it seems like there is some dispute over whether mass increases with speed or momentum increases with speed. In the end, though, it has the same physical effect. The momentum of an object is given by:
p = mv / (sqrt( 1 – (v^2 / c^2) ))
Where m is the rest mass, v is velocity (relative to the frame of reference), and c is the speed of light.

At low speeds, this is the same as the traditional formula for momentum. When you approach the speed of light, however, the momentum increases rapidly, approaching infinity as v=c.

Thanks for responding, @PhiNotPi. It is my understanding that Observer A would “see” B as moving slower and slower as B approached the event horizon, while B’s experience of its own time would be “normal.” B’s observation of the black hole would be that the approach was slowing (along with the rest of the universe). Once the event horizon was reached, progress for B would seem to have stopped, and B would wait forever while its time continued to pass. Am I correct in this?

As I began writing this you added the comment regardling speed/momentum. Being somewhat math challanged, I will have to opt out of that part of the discussion for now. The notion that even a proton’s mass begins to approach infinity as it accelerates to near light speed, since mass begets gravity, and given the amount of steller material that falls into a black hole, its gravitation force should eventually become unlimited (which, of course, it does not). I look forward to your thoughts.

I’m not sure that the added mass adds more gravity. The rest mass of the particle does not increase, but the relativistic mass is what does. The whole idea of relativistic mass is sometimes rejected due to its confusing nature. Einstein himself said that it is better to think of it as additional momentum, not additional mass.

I’m researching this as I’m answering (I am not an expert in this field, or any field), and I don’t think that the additional mass actually creates more gravity, since the rest mass is always the same.

@Bill1939 Great question. I am no expert either, but this is a question I’d love to bounce off one of the Ask a Physicist sites. NASA’s Ask an Astrophysicist site would be a good resource. I would encourage you to post the question there with a link to this discussion and see what they have to say.

My tendency is to agree with @PhiNotPi that the proton’s momentum and not mass would approach infinity. Another interesting aspect of this is what happens to it inside the event horizon. With near infinite momentum, what does it do to whatever massive object it collides with at the singularity? It doesn’t seem likely that it could pass straight through without encountering anything and sail back out toward the event horizon on the other side of the black hole before the supper-massive gravitational well pulls it back in. But we are in an area of phenomena where physicists currently just have to speculate, as all that’s beyond the event horizon is unobservable to us and thus a mystery.

As far as I can tell, most theories, hypotheses actually, about black holes are conjectures, albeit by highly intelligent and educated physicists and mathematicians. I postulated the possibility on the Live Science website that nothing existed inside an event horizon, and that a black hole’s gravitational force was derived from the mass that accumulates at the horizon. No one even bothered to acknowledge the question. Either they rejected it out of hand or it was too stupid (probably the latter).

As my bio shows, I am not an expert at anything. My IQ, as measured by the military in 1960, was 125. Sixty years later, it is likely considerably less. I keep trying to wrap my head around time dilation, but only understand (I think) why gravity effects time as it does. Part of my problem is an unwillingness to accept that no form of time is universal. Some day I will elaborate on this point.

I think I understand why/how momentum might be a better fit in E=MC^2 than mass. The notion that mass could become infinite, yet not fill all of space seems credulous (although compressing an infinite mass to a singularity seems almost okay). I can see that momentum would not be related to gravity, while mass almost certainly must be (unless there can be matter without gravitons).

I will pass on sending this question to NASA’s Ask an Astrophysicist site, @ETpro, until I have a better understanding of the issues involved. Mathematical equations go over my head, and unless they are willing to give an answer that a twelve-year-old would understand, such efforts on my part would be futile.

You don’t necessarily accelerate to relativistic velocity just before falling through a black hole’s event horizon. Accelerating to near-light speed is what it would take to escape from just outside of it. These are two quite different things.

What you say, @hiphiphopflipflapflop, is true, but I am not sure how it relates to what I wrote. An object, in this case a proton, being drawn by a black hole for a sufficient length of time to have accelerated to near light speed just before reaching the event horizon, cannot escape. (I should have spelled out this scenario before. I guess I assumed it to be understood. I apologize for my lack of forethought.) Before @PhiNotPi suggested that at relativistic velocity momentum and not mass was accelerating, I imagined that the mass would envelope a black hole at its event horizon, and would not penetrate and reach singularity.

@Bill1939 I’d like to see if we’d get any answer. Will you allow me to Ask a NASA Astrophysicist?

Here’s how it may relate: what I have read about black holes makes me think that an object falling in “from infinity” into a particularly massive black hole (such as those believed to be present in the center of many galaxies) might not be traveling relativistically compared to its original inertial frame “at infinity” when it crosses the event horizon. I’m not sure about this. I haven’t been able to find an equation that describes the velocity of an object falling into a gravitational potential well from infinity. I did come across this site though, and you may find it interesting though it may not answer your question.

Be my guest, @ETpro. Please share whatever Ask a NASA Astrophysicist responds.

@hiphiphopflipflapflop, you may have a better grasp of time dilation due to relative velocity than I do.

A proton that falls into a black hole will only add the mass of one proton to the black hole however fast it is travelling.

From the point of view of the proton nothing appears to change as it crosses the event horizon. Observer B who is travelling alongside the proton will continue to observe the universe as before unaware that he has passed the point of no return.

From the point of view of observer A the proton never reaches the event horizon as time slows down so much that the proton would take an eternity to get there.

What is interesting about black holes is that they are subject to the laws of quantum mechanics and of general relativity, two mutually incompatible theories. We will only understand black holes when we can unite these two theories into one.

Thanks, @flutherother. I had long been given to understand E=MC^2 to mean that a mass increased as its velocity approached light speed, and that at light speed the amount of energy required would be infinite and the mass would be infinite. I gather that this is not true.

@Bill1939 E=MC^2 is the formula for the amount of energy released when matter is converted into energy. In the current form, it applies mainly when that particle is at rest or is traveling at classical speeds.

If the particle is traveling, it’s relativistic mass is given by the formula
M= m / (sqrt( 1 – (v^2 / c^2) ))

As an object approaches light speed, it is perfectly acceptable to say that its relativistic mass approaches infinity. If you have a ping pong ball that is traveling fast enough, it will have the same momentum as a freight train. However, the additional mass does not add any gravity, which is why it is good to think of it as momentum.

Technically, whether is it mass or momentum that is increasing, the math works out to be the same, so it is really only an a debate over terminology.

Again, thanks @PhiNotPi. I see that it is the kinetic energy of a mass and not the physical mass that increases with increased velocity. It’s obvious, now that you have pointed it out to me.

@Bill1939 I thought the same as you, but it would mean that objects travelling close to the speed of light would become black holes. This doesn’t happen. The actual mass of an object as measured by an observer who is travelling with it will not change. The mass that a stationery observer will observe does increase with speed and is called ‘relativistic mass’ though how this can be measured I can’t imagine.

@flutherother, you say that the “mass that a stationery observer will observe does increase,” but are you suggesting that the observed object will appear to become larger? A bullet with a half load of gun powder will strike a target with less force than one with a full load, though the mass of the bullet in both cases is the same. If we could use enough gun powder to propel the bullet near light speed, would its mass change? I think @PhiNotPi has it right. It is not the physical mass that changes, but the kinetic energy, the momentum of the mass that changes.

It is possible for relativistic mass to increase without the size increasing. The best way I can describe it is that the density increases.

If the size did not change, how would an observer know that the mass had changed (besides mathematically)? If the density increases, wouldn’t any object reach singularity at light speed (it’s hard to imagine a proton becoming a black hole)? Until someone suggests an explanation of mass increasing without increasing size or density, I am going to stay with my newly found belief in the substitution of momentum for mass in Einstein’s equation. Kinetic energy seems a very reasonable answer to this conundrum for me at this time.

@Bill1939 The question is in. Let’s see what the volunteers at NASA have to say—of they choose to say anything.

@Bill1939 WooHoo! I did get an answer. Hans Krimm of NASA wrote:

You and your group are right. In a more rigorous study of relativistic effects, it is not really correct to talk about mass increase with speed. The more proper relativistic treatment is to look at total energy, E, which is related to rest mass m and momentum p by the equations

E^2 = (mc^2)^2 + (pc)^2
E = gamma*(mc^2)
p = gamma*(v/c)*mc

where the Lorentz factor, gamma is 1/sqrt(1—(v/c)^2)
as v->c, gamma->infinity as does p. Since the rest mass is invariant, a larger and larger portion of E is in momentum.

Cheers,

Hans Krimm for Ask an Astrophysicist

So there we have it. @PhiNotPi answered correctly.

Thanks, @ETpro. However, I still somewhat confused. Hans Krimm said @PhiNotPi answered correctly. But @PhiNotPi said, “It is possible for relativistic mass to increase without the size increasing. The best way I can describe it is that the density increases.” Is it density or momentum that accounts for the increase?

I’m not so sure that I got it right (Hans Krimm did not actually say that I was correct). But it seems like the best way to describe it is not mass increasing, but the total energy increasing.

Thanks @PhiNotPi. I understand that the mass does not actually change. The energy of its momentum accounts for the increase in the apparent mass of the proton that mathematics shows.

@Bill1939 That’s my understanding of what Hans Krimm is actually saying, and demonstrating with his maths.