Mass & Acceleration

[WestAir]

Really quick question:

If accelerating an object to relativistic speeds increases its mass, and there is no limit to how fast you can accelerate an object (because infinity is the limit), then: Is it possible to accelerate an object so fast that its mass increases to the point where it is massive enough to fall within its own event horizon and become a black hole? If so: Is this a local phenomenon (only apparent from a nearby observers POV) and if so, how does relativity handle the discrepancy of what is or isn't within its own event horizon?

[GregoryNeal]

The notion of relativistic mass has fallen out of favor with the physics community. When physicists talk about mass, they are talking about the rest mass. I have a couple of links for you to read up on if you want to find out more, as I have no great knowledge of GR. However, I do know that you cannot just replace the mass from Newtonian Mechanics with the relativistic mass when talking about an objects gravitational field.

http://www.physicsforums.com/showthread.php?t=144817

http://www.physics.adelaide.edu.au/~dkoks/Faq/Relativity/BlackHoles/black_fast.html

http://en.wikipedia.org/wiki/Mass_in_special_relativity#Transverse_and_longitudinal_mass

[K^2]

Yeah, it's a little bit more complicated. The actual source of gravity is the stress energy tensor. Energy increase, which is proportional to relativistic mass increase, is going to factor into it. But so will momentum. For a point mass, it is a quantity proportional to the outer product of object's four-momentum with itself. The resulting gravitational field is a Lorentz boost of the original field (precisely because you can simply look at it from a moving coordinate system), so if the object didn't have an event horizon before you started to accelerate it, it won't have it even once you've gotten the object to relativistic speeds. In other words, you can't make a black hole by simply taking an object and accelerating it.

[WestAir]

K^2,

Thanks for the explanation. While you're still here, would you mind explaining to me why my slightly off-topic thought experiment won't work? Here's the jist, from another thread:

Question for the science professionals here:

What is the difference between an object with mass traveling at the speed of light (299,792,458 m/s), and an object traveling at almost the same speed, but slower by 1 Plank Length x a time larger than the age of the Universe? One is unachievable, and the other is achievable, yet both would have the object move the exact same distance over the exact same time if clocked today. How is that explained?

Sorry westair, got my degree nearly 10 years ago and don't remember anything about that. To be honest I'm not sure what you are asking, can you reference an article or be more specific?

Speed is determined by distance x time. In this case, meters x seconds. The plank length, too, is a distance. The difference here is that you can't crunch any information smaller than the plank length - so something that's moved 0.1 Plank Lengths is in the exact same physical position as something that has moved 0.6 Plank Lengths. There's no difference. There is a similar variable for time; if it takes you "the age of the universe" to move 1 meter, then you've moved 1 meter as of today. If it takes you "longer than the age of the universe" to move 1 meter - well you've yet to move 1 meter.

The thought experiment here is that there is a physical, non-infinite energy requirement for an proton to travel 10 plank lengths in 10.5 plank times. However, because you can't fractionate the plank length, both the proton and a photon will have traveled the same physical distance. The thought experiment can be dragged even deeper. Pretend both a proton and a photon began their journey during the birth of the Universe. If the protons speed is slower than a photon by 1 plank length x the age of the universe plus one second, then up until now, it's kept pace with a photon - it's traveled at C, which is impossible, but will become possible in the future when it begins to lose the race.

How can that be without breaking the rules? Surely the energy requirements needed to accelerate the proton to this speed are calculable, and surely some Oh My God particle has been traveling at this speed somewhere. It's NOT traveling at the speed of light, but because of the existence of the Plank Scale, it should appear to be.

WestAir, I might also note that you're trying to take a QM approach to Relativity (Plank Length is a QM concept), and as we know, the two do not coexist peacefully yet. So it may be that no one truly knows the answer anyway.

It takes an infinite amount of energy to accelerate a particle with non-zero rest mass up to the speed of light. What you are talking about would require such a tremendous expenditure of energy into a single particle that I assure you, no such particle exists in the universe. It would probably have the energy of like a, AT LEAST a large hydrogen bomb, maybe something closer to a supernova, packed into a single particle. There is just no way, natural or otherwise, to focus that kind of energy into a single particle. I tried to calculate the amount of energy involved, but my computer is simply not capable of calculating down to that many decimal places. Maybe MATLAB could, but I doubt it.

Anyway, I figured out a way to do the calculation, and it turns out, if my numbers are right the particle would have an energy of something like 3x10^20 J. That is over 1500 times the energy of the largest hydrogen bomb ever tested. It's equivalent to 3600 kg of mass. I have to wonder whether that is enough mass, packed into the size of a proton, for the particle to collapse into a mini black hole. If so, would it? From the frame of reference of the particle, its mass doesn't increase, or does it? From our frame of reference, it should, and it might collapse under its own gravity. I'm not sure how this is resolved, I've never thought about the mass increase at relativistic velocities much. If it did collapse, it would rapidly dissolve in a burst of Hawking radiation.

Finally, note that your hypothetical particle is over 6x10^18 times more energetic than the most energetic cosmic ray EVER detected. Such a thing does not exist in this universe, you can be sure of that.

Edit: Just to make sure we're on the same page, this is the speed I assumed for the proton:

One plank length behind a photon after the photon and proton had traveled 15 billion light years.

That's about one part in about 10^61 less than the speed of light. I had a hard time doing the calculation till I wrote it out and realized that (1 - (1-10^-61)^2)^0.5 could be simplified to (2*10^-61)^0.5.

It sounds like, if QM and Relativity could be assumed to flesh out evenly with this experiment, that 3x10^20 Joules is the single energy requirement below a value of infinity [today], and adding more energy would do nothing to accelerate the proton further [until the future when spacetime expands further]. It also means that the energy required to reach "a quanta below infinity" changes based on the distance being measured, and you can only reach true infinity with a distance of infinity...

Unless I'm dead wrong, which is really what I'd like to know from the above wall-o-text.

[K^2]

We don't actually know how things behave at Plank scales. We think they behave differently, but it doesn't necessarily mean that things are going to come quantized in units of Plank length, for example.

Edit: Relativity is fairly straight forward. So is Quantum Mechanics when you get right down to it. Either one is a pain to actually make any computations in, except for some very simple cases, but conceptually, they are very compact and consistent. So if you ask a question about either one, I can probably give you a straight answer. If not quantitative, then at least qualitative. But when you start mixing the two... Quantum Gravity is not renormalizable, even if you manage to formulate it, and non-quantum Unified Field Theory, while entirely possible to formulate, is pretty much impossible to make any use out of. So in almost any situation where you mix the two, things just break down, and it becomes hard to answer even the most basic questions about what may and may not be.

Edited October 13, 2013 by K^2

[WestAir]

Well that's discerning, but at least enlightening.