# Relativistic mass increase appearing to form a black hole vs. a real black hole

As I understand it, if you are moving fast enough relative to me, your mass increase will cause you to APPEAR like a black hole to me, even though you will not see yourself as a black hole. It would be an illusion, right? And also you would see me as a black hole while I would reject that view.

Ok, so how do we know the same thing isn't happening in some way with a "real" black hole? Maybe a black hole only APPEARS to be a black hole when it's really not.

atyy
A point particle in classical GR is a black hole (not an illusion).

You, however, are not a point particle in classical GR. You are a collection of fields. These fields have stress-energy-momentum. Like the relativistic mass, these have different values in different spacetime coordinate systems. However, the values always change such that whether spacetime and matter contain a black hole has the same answer in all coordinate systems.

A point particle in classical GR is a black hole (not an illusion).
How can you even argue a (neutral) point particle in classical GR can have any mass?

atyy
How can you even argue a (neutral) point particle in classical GR can have any mass?

OK, I mean mass parameter (like the Schwarzschild solution). A point particle would have a radius less than its Schwarzschild radius, so a reasonable interpretation would be that it's a black hole.

PeterDonis
Mentor
OK, I mean mass parameter (like the Schwarzschild solution). A point particle would have a radius less than its Schwarzschild radius, so a reasonable interpretation would be that it's a black hole.

A true "point" particle would, yes, but when you get down to those distance scales you have to take into account quantum effects and the possibility that what we think of as point particles, like electrons, are actually extended objects (for example, string theory takes this view). If the distance scale at which this happens is the Planck length or thereabouts, as many quantum gravity theorists seem to think, then it is *not* true that our typical "point particles" will have a radius less than their Schwarzschild radius, because the mass of a black hole with Schwarzschild radius equal to the Planck length is the Planck mass, which is about 10^-5 grams, far larger than the mass of any particle we might think of as a point particle.

Also, even particles like electrons that are thought of as "point particles" in some contexts are thought of as having finite radius in other contexts. For example, the electron's Compton wavelength, which is one common candidate for its "radius", is about 10^-13 m IIRC, far larger than its Schwarzschild radius.

Chronos
Gold Member
A mass that is not a black hole in its own reference frame is not a black hole in any reference frame, period.

A mass that is not a black hole in its own reference frame is not a black hole in any reference frame, period.

I guess that's part of what I'm asking. How do we know that it is a black hole in it's own frame of reference? How do we know that what we think of as it's frame of reference isn't wrong?

atyy
I guess that's part of what I'm asking. How do we know that it is a black hole in it's own frame of reference? How do we know that what we think of as it's frame of reference isn't wrong?

The relativistic mass and frame of reference are concepts from special relativity. In special relativity, one can have point particles. There is no gravity, so there is no question of a point particle being a black hole.

In general relativity, a point particle is best thought of as a black hole. As PeterDonis says, electrons are not point particles in this sense. Because spacetime is curved, it is not very helpful to think of the frame of reference of a black hole. Let's take the simplest black hole - the Schwarzschild solution. There are many coordinate systems describing this black hole spacetime. A black hole is defined in all of them as a surface called the event horizon beyond which an object inside cannot communicate with an object outside. The definition of an event horizon doesn't depend on the coordinate system we choose. How do we know an object is a black hole? General relativity predicts that spacetime outside the event horizon has particular spacetime features. Although much evidence indicates that there are black holes at the centres of galaxies, we don't know for sure. There are proposals to falsify the idea by using infalling objects and the radiation they emit to probe those spacetime features.

Last edited:
pervect
Staff Emeritus
One of the predictions of Special Relativity is that there is no way to determine an absolute velocity. It's possible to think of theories where one could measure absolute velocity, but these theories are not SR. There's also a fair number of failed attempts to measure absolute velocity, such as the Michelson Morley's experiment's failure to find any "ether wind", so we have some reason to believe that SR is correct on this point based on experiments to date. If we ever do find a way to measure absolute velocity, it will falsify SR.

Because GR is built on top of SR, and GR is basically a statement that SR works locally, this is also a prediction of GR.

Given that you can't simultaneously be a black hole and at the same time not be a black hole, and also given that you can tell when something is a black hole by experiment, it should be obvious that the principle that there isn't any way to determine absolute velocity via measurement rules out the possibility that you turn into a black hole sometimes and not others depending on your "state of motion".

Some miscellaneous points:

1) Netwon's force law doesn't work at all for fast moving objects. In particular, there isn't any spherical symmetry to the field of a moving object.

2) Newton's force law can't work, even in priniple without being modified for fast moving objects, because it's not covariant. So point #1 is really expected, not a surprise. The necessary relativistic corrections for electromagnetic forces turn out to be magnetism. There are similar corrections for gravity.

3) Studying the force law of an electric charge gives you some idea of how a rather similar force actually behaves relativistically. It's not quite identical to gravity, but it'll get you a lot closer than using Newton's laws, which just won't work. It's also easier to do, and is covered in most E&M textgbooks. Going into details is interesting, but would make this post too long and start to go off the point, but ask and/or start another thread if you're interested.

4) The concept of "relativistic mass" turns out to be a dead end. It's not particularly useful in computing gravity. So not only is Newton's law of gravity the wrong law, "relativistic mass" is the wrong mass to use in the law, once you get the correct law. In general, one needs to use the stress-energy tensor (which is a matrix of values, not a single number) to compute gravity. Under special circumstances, where one has a static geometry (this rules out moving masses, by the way!) one can use the Komar mass, which is a scalar, rather than the more complex stress-energy tensor for this purpose. Howeer, the Komar mass isn't the same as the "relativistic mass" from Special Relativity.

5) Measuring gravity is a bit trickier than it looks. The best way of doing it is to measure tidal gravity. Without a gravitationally neutral object, which doesn't exist, you can experimentally determine geodesic motion, but the "force" of gravity for something undergoing geodesic motion is always zero, which isn't particularly useful in defining a "force". You can avoid this issue most easily by measuring the rate of change of the force, i.e. the tidal force, which is very closely related to the Riemann tensor.

This should be enough, or more than enough, for now

Last edited: