A question about the consequences of relativity.

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opsomath
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A question about the consequences of relativity.

Postby opsomath » Mon Jun 09, 2008 3:21 pm UTC

So, I understand that under special relativity, the relativistic mass of an object varies proportionally to the Lorentz factor. (inversely to (c^2-v^2))

Furthermore, for a given mass, it is possible to calculate the Schwartzchild radius, ie, if you squish it into a small ball of this radius, you have made a black hole. (Congratulations.) This formula can be inverted, though, to calculate a mass needed for an object of a given radius to be a black hole.

Therefore, I have a thought experiment.

Take a proton. You know the radius, or an upper bound on it, as well as the mass. Calculate, using these values and the Schwartzchild formula, a velocity. Accelerate it to that velocity using a massive electrical field.

Does it become a black hole?

Furthermore, black holes have charge. (Wikipedia says so.) So, use a second massive electrical field to decelerate it down below this critical speed. Does it stop being a black hole?

Next question. Wikipedia says that many people think black holes contain no information other than mass, charge, and angular momentum. Say we perform our experiment with any other species than a proton; for instance, He+. This species contains an electron, and thus has an electronic state. This is information. What happens to that information during our experiment? If the experiment is possible, it would shed light on the "black hole information paradox" that the Wiki also mentions.

Look forward to hearing everyone's responses.

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Re: A question about the consequences of relativity.

Postby Robin S » Mon Jun 09, 2008 4:12 pm UTC

I am not a physicist, but I was under the impression that only rest mass affected whether or not an object was a black hole.
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Re: A question about the consequences of relativity.

Postby opsomath » Mon Jun 09, 2008 4:30 pm UTC

Robin S: Yeah, I'm afraid only a "real physicist" is going to be able to answer this one definitively. Is the apparent gravitational mass not affected by the velocity, only the apparent inertial mass? If so, that brings up an interesting point in itself.

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Re: A question about the consequences of relativity.

Postby seladore » Mon Jun 09, 2008 6:11 pm UTC

The apparent mass would have to be affected by the velocity.
Otherwise, you could look at your proton that should be a black hole, see it isn't, and deduce that you are travelling at a constant (high) velocity. It violates relativity, in other words.

Actually, as I write this I realise that it is wrong. The proton would have rest mass according to you if you were in the same frame. The question seems to be, "can an object be a BH in one frame, and not in another".

That is really interesting, and I don't know the answer.

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Re: A question about the consequences of relativity.

Postby danpilon54 » Mon Jun 09, 2008 6:38 pm UTC

I am nearly 100% sure that relativistic mass does not add to gravitational mass. I don't like the analogy that objects gain mass at high speeds. Instead they gain inertia. It is harder to accelerate an object traveling near c, but its gravity is the same.
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Re: A question about the consequences of relativity.

Postby seladore » Mon Jun 09, 2008 7:01 pm UTC

But I thought that the whole point of relativity was that inertial mass is equivalent to gravitational mass?

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Re: A question about the consequences of relativity.

Postby nilkemorya » Mon Jun 09, 2008 7:05 pm UTC

Question1: My first modern physics prof. used this for relativistic effects, if it happens in one frame, it happens in all of them. It might not happen in the same way at the same time, but it happens in all frames, the timing will be different, but all observers should be able to agree on what events happened. So I can't see it turning into a black hole.

Question2: I think with the "no black hole" answer above this is answered. So, charged black holes are really unlikely as far as I can tell, and even if they are charged, I don't understand how to determine that charge, since it seems that the charge is still within an event horizon... Anyone?
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Re: A question about the consequences of relativity.

Postby danpilon54 » Mon Jun 09, 2008 7:26 pm UTC

seladore wrote:But I thought that the whole point of relativity was that inertial mass is equivalent to gravitational mass?


I believe it's that rest mass is equivalent to gravitational mass, otherwise, like others have said, it would violate the principle of relativity because in one frame, the mass would collapse into a black hole, and in the other, it would not.
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Re: A question about the consequences of relativity.

Postby EricH » Mon Jun 09, 2008 7:54 pm UTC

nilkemorya wrote:Question2: I think with the "no black hole" answer above this is answered. So, charged black holes are really unlikely as far as I can tell, and even if they are charged, I don't understand how to determine that charge, since it seems that the charge is still within an event horizon... Anyone?


Charge is a property of the whole hole...er, sorry. The electric field from a charged black hole is, AFAIK, indistinguishable from that of a point charge at the singularity--at least, indistinguishable outside the event horizon. How to determine that charge? Apply an electric field, or a changing magnetic field, and measure the black hole's acceleration.
Just because a black hole is involved doesn't mean all of Newtonian physics is invalid....
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Re: A question about the consequences of relativity.

Postby vanRillandBath » Mon Jun 09, 2008 9:23 pm UTC

I think saying that the mass of an object increases when you speed up is a bad way to look at it. The rest mass stays the same. However, in your equations a factor gamma shows up. This causes the extra inertia. For example, instead of [imath]E=mc^2[/imath] you should write [imath]E=\gamma mc^2[/imath], where [imath]\gamma = 1/\sqrt(1-v^2/c^2)[/imath].

So you cannot replace [imath]m[/imath] with [imath]\gamma m[/imath] in the formula for the Schwarzschild radius ([imath]r_s = \frac{2Gm}{c^2}[/imath]). That formula is correct as it stands.

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Re: A question about the consequences of relativity.

Postby danpilon54 » Mon Jun 09, 2008 11:13 pm UTC

vanRillandBath wrote:I think saying that the mass of an object increases when you speed up is a bad way to look at it. The rest mass stays the same. However, in your equations a factor gamma shows up. This causes the extra inertia. For example, instead of [imath]E=mc^2[/imath] you should write [imath]E=\gamma mc^2[/imath], where [imath]\gamma = 1/\sqrt(1-v^2/c^2)[/imath].

So you cannot replace [imath]m[/imath] with [imath]\gamma m[/imath] in the formula for the Schwarzschild radius ([imath]r_s = \frac{2Gm}{c^2}[/imath]). That formula is correct as it stands.



Correct in principle, but not in example. E=mc^2 is the formula for the rest mass energy of an object, while [imath]E=\gamma mc^2[/imath] is the formula for the total energy of an object.

[imath]\gamma[imath] is a result of special relativity, which does not deal with gravity. This does not necessarily mean some results of SR are not valid in GR, but one cannot assume that all results such as increased "mass" due to velocity, hold.
Last edited by danpilon54 on Mon Jun 09, 2008 11:25 pm UTC, edited 1 time in total.
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Re: A question about the consequences of relativity.

Postby Owehn » Mon Jun 09, 2008 11:24 pm UTC

danpilon54 wrote:
vanRillandBath wrote:I think saying that the mass of an object increases when you speed up is a bad way to look at it. The rest mass stays the same. However, in your equations a factor gamma shows up. This causes the extra inertia. For example, instead of [imath]E=mc^2[/imath] you should write [imath]E=\gamma mc^2[/imath], where [imath]\gamma = 1/\sqrt(1-v^2/c^2)[/imath].

So you cannot replace [imath]m[/imath] with [imath]\gamma m[/imath] in the formula for the Schwarzschild radius ([imath]r_s = \frac{2Gm}{c^2}[/imath]). That formula is correct as it stands.



Correct in principle, but not in example. E=mc^2 is the formula for the rest mass energy of an object, while [imath]E=\gamma mc^2[/imath] is the formula for the total energy of an object.


I think vanRillandBath properly explained it: "E=mc^2" is the oft-quoted equation for relating the (total) energy of a particle to its so-called "relativistic mass". ("As an object speeds up, its energy increases, so its mass increases as well, making it harder to accelerate" is the popular line of reasoning.) VanRillandBath is roughly saying "If you reformulate everything involving 'relativistic mass' in terms of rest mass, then E=mc2 becomes [imath]E=\gamma mc^2[/imath]," which is correct.

Edit: Whoops, mixed up who I was talking to.
Last edited by Owehn on Mon Jun 09, 2008 11:35 pm UTC, edited 1 time in total.
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Re: A question about the consequences of relativity.

Postby danpilon54 » Mon Jun 09, 2008 11:28 pm UTC

Hmm yea I guess thats just a different way of looking at it. [imath]E=mc^2[/imath] is the total energy of a particle at rest, since [imath]\gamma[/imath] is 1. My bad, but I think that way of thinking is slightly confusing, as you plug in for 1 variable but not the others.
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Re: A question about the consequences of relativity.

Postby CFT » Tue Jun 10, 2008 7:59 am UTC

1. In general relativity, the gravitational field equations include more than just the mass of an object. They include the energy momentum tensor, which is a 4x4 matrix. This matrix has the energy of the object as one of it's components.

2. The Schwarzschild solution only refers to a stationary black hole. Kerr solved for a rotating black hole. There is also a solution for a charged black hole. I don't know if you can apply the Schwarzschild radius for a black hole rotating at relativistic speed.

3. In the sizes you refer to, quantum mechanical effects come into play. Your black hole will probably evaporate very quickly due to hawking radiation.

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Re: A question about the consequences of relativity.

Postby opsomath » Tue Jun 10, 2008 10:49 am UTC

CFT wrote:1. In general relativity, the gravitational field equations include more than just the mass of an object. They include the energy momentum tensor, which is a 4x4 matrix. This matrix has the energy of the object as one of it's components.

2. The Schwarzschild solution only refers to a stationary black hole. Kerr solved for a rotating black hole. There is also a solution for a charged black hole. I don't know if you can apply the Schwarzschild radius for a black hole rotating at relativistic speed.

3. In the sizes you refer to, quantum mechanical effects come into play. Your black hole will probably evaporate very quickly due to hawking radiation.


Thanks for your informative reply!

So, I am to infer from 1) that yes, the kinetic energy of the object does in fact contribute to its gravitational field. Concerning 2), I did not envision a rotating object, but essentially a proton (or more complex ion) in a linear accelerator or a storage ring. However, it's interesting that the solution is different for a charged hole. Unsurprisingly, I didn't uncover a description of this on a brief Wikipediaing. How would this change the situation?

Concerning 3), I wondered about this. Since the Wiki article on Hawking radiation seems to consider masses about the size of the Moon, which contains more than one proton, and suggest that sufficiently small black holes annihilate themselves in a burst of gamma rays, I infer that a proton-sized black hole would do so immediately. However, where would the charge go?

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Re: A question about the consequences of relativity.

Postby Owehn » Tue Jun 10, 2008 11:21 am UTC

1) Not as such. If you have a proton traveling at high speed, you can always boost to its rest frame, in which it won't seem to be moving at all. (You can think of it this way: The relevant qualities of a particle are its energy and momentum, and any relativistic quantity that depends on the energy must depend on the momentum, and vice versa. But if a relativistic scalar quantity depends on a four-vector like (energy, momentum), it must depend only on the squared length of the four vector, which for a particle is always just the squared rest mass, i.e. [imath]m^2=E^2-p^2[/imath].) However, if you had two boxes, one with a hot gas full of speedy particles, the other with a cool gas of slow particles, the hot gas would have a slightly stronger "gravitational pull" and in that respect be more susceptible to conversion into a black hole.

2) The reason the solution is different for a charged black hole is rather subtle: the electromagnetic field the hole induces permeates the surrounding area, and itself contributes to the stress-energy tensor, so the space is warped differently.

3) I'd guess that the charge of the black hole would manifest itself in a slight bias in the charges of the radiated particles, so that as the hole loses mass it would also lose charge to the rest of the universe.
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Re: A question about the consequences of relativity.

Postby thoughtfully » Tue Jun 10, 2008 2:06 pm UTC

seladore wrote:But I thought that the whole point of relativity was that inertial mass is equivalent to gravitational mass?


Only locally. Zoom out a bit and you can distinguish them with tidal forces and such.
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Re: A question about the consequences of relativity.

Postby nilkemorya » Tue Jun 10, 2008 3:19 pm UTC

Thanks eric, can't believe I forgot that one. I don't know how many times I heard "How do you know if you have a charge? Put it in an electric field and see if it feels a force." Or the opposite "Take a point charge and let it go, does it move? Then you have an electric field." I gotta stop posting in the morning before my brain works.
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Re: A question about the consequences of relativity.

Postby opsomath » Tue Jun 10, 2008 3:36 pm UTC

[quote="Owehn"

3) I'd guess that the charge of the black hole would manifest itself in a slight bias in the charges of the radiated particles, so that as the hole loses mass it would also lose charge to the rest of the universe.[/quote]

Okay. But what particles would be radiated? Gamma rays do not possess charge. Surely it wouldn't be just a less energetic proton?

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Re: A question about the consequences of relativity.

Postby Owehn » Wed Jun 11, 2008 12:36 am UTC

A proton, or a positron, or a W+, or 500 protons and 499 electrons. There isn't anything really special about gamma rays that causes them to be emitted by evaporating black holes, it's just our word for energetic photons. Other particles are bound to be mixed in, but possibly in smaller quantities and not as easily detectable.
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Re: A question about the consequences of relativity.

Postby antonfire » Wed Jun 11, 2008 2:04 am UTC

There's a bit at the end of this thread about it.

Quick summary: things moving quickly don't make black holes, even though they exert more "gravitational pull" than they would if they were at rest. In your frame, it looks like the object is more energetic, and pulls you harder. In the object's frame it looks like your time is slowed so you seem to observe a greater acceleration. "Gravitational pull", "energy", "relativistic mass", and so on are all frame-dependent things. Whether something is a black hole is frame-independent.

Metasummary: It doesn't become a black hole because relativity is more complicated than that.
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Re: A question about the consequences of relativity.

Postby gravitysrainbow » Wed Jun 11, 2008 5:33 am UTC

antonfire wrote:There's a bit at the end of this thread about it.

Quick summary: things moving quickly don't make black holes, even though they exert more "gravitational pull" than they would if they were at rest. In your frame, it looks like the object is more energetic, and pulls you harder. In the object's frame it looks like your time is slowed so you seem to observe a greater acceleration. "Gravitational pull", "energy", "relativistic mass", and so on are all frame-dependent things. Whether something is a black hole is frame-independent.

Metasummary: It doesn't become a black hole because relativity is more complicated than that.



thanks for putting that in here!

also:
seladore wrote:But I thought that the whole point of relativity was that inertial mass is equivalent to gravitational mass?


That is one of the foundational ideas (aka the equivalence principle). So, you're on to something important, but that doesn't really fit in here because you can only measure inertial mass and "gravitational mass" in the same frame. So, the idea of "increasing the mass" by putting an object in another frame will never create a black hole, because _in the frame of the object_ nothing is changing.

Cheers,

GR


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