## Main Question or Discussion Point

Hi all - this (seeming) paradox has been brought up on a few other forums with less than illuminating results. I'm curious to see what the solution is.

Okay, so, you're watching a massive object approach the speed of light. From your reference frame, you are measuring the increase in apparent mass due to relativistic effects. This thing goes faster and faster and gains more and more energy, and so appears to get more and more massive. At some point, this thing will appear to be so massive that it should collapse under its own gravitation to form a black hole. However, from *its* reference frame, its apparent mass is constant - just its rest mass - and so clearly no black hole forms. That is, from one reference frame this thing collapses, while from the other reference frame it does not: paradox.

How is this reconciled? Where am I going wrong?

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The problem is your concept of mass. You are using "resistance to acceleration" as your definition. It is true that the "resistance to acceleration" one would feel trying to accelerate an object faster and faster would increase with speed.

However, mass (especially if you want to consider gravity, which is a whole 'nother issue) should not be considered this way. Newton's second law is:
F = dp/dt.

What you are really seeing with "increased mass" is that momentum is not proportional to velocity as it is for non-relativistic mechanics. Instead:

p = m v / Sqrt( 1 - v^2 / c^2)

The mass of the object however stays constant. What I am using to define mass, as is the common definition physicists use, is the invarient mass. Which is also equal to the "rest mass" of a system.

Does that help some?

As a follow up thought experiment to help:

If we use "resistance to acceleration" as the definition of mass, try writing out the equations for "mass" if you push a moving object A) in the direction it is already travelling or B) prependicular to the direction it is travelling.

You will find A and B don't give the same answer!
This is but one reason physicists don't use that concept to define mass.

Dale
Mentor
Hi all - this (seeming) paradox has been brought up on a few other forums with less than illuminating results. I'm curious to see what the solution is.

Okay, so, you're watching a massive object approach the speed of light. From your reference frame, you are measuring the increase in apparent mass due to relativistic effects. This thing goes faster and faster and gains more and more energy, and so appears to get more and more massive. At some point, this thing will appear to be so massive that it should collapse under its own gravitation to form a black hole. However, from *its* reference frame, its apparent mass is constant - just its rest mass - and so clearly no black hole forms. That is, from one reference frame this thing collapses, while from the other reference frame it does not: paradox.

How is this reconciled? Where am I going wrong?
Hi Greg, welcome to PF.

You should look at the Pysics FAQ page entitled http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/black_fast.html" [Broken]. The bottom line is that the black hole is a feature of the Swartzschild solution to the Einstein field equation. That solution assumed a stationary and non-rotating spherical mass, so it doesn't apply to a rapidly moving mass.

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Look at it this way:

Two cannon balls are separated by a short distance. Far from any other massive body they fall towards each other in time (t) due to the mutual gravity of their own masses as measured by an observer at rest with the zero momentum frame of the two balls.

To an observer moving at relatavistic speed tranverse to a line joining the centres of the cannon balls, the cannon balls move together much slower due to time dilation. This observer considers himself to be stationary and the cannon balls are moving at relatavistic speed according to him. So the cannon balls appear to have LESS mutual gravitational attraction (because they accelerate slower towards each other) and NOT MORE when moving at relativistic speeds. When a body is moving at relativistic speeds it's constituent particles have less mutual gravitaional attraction and therefore does not become a black hole in any inertial reference frame. :)

The problem is your concept of mass. You are using "resistance to acceleration" as your definition. It is true that the "resistance to acceleration" one would feel trying to accelerate an object faster and faster would increase with speed.

However, mass (especially if you want to consider gravity, which is a whole 'nother issue) should not be considered this way. Newton's second law is:
F = dp/dt.

What you are really seeing with "increased mass" is that momentum is not proportional to velocity as it is for non-relativistic mechanics. Instead:

p = m v / Sqrt( 1 - v^2 / c^2)

The mass of the object however stays constant. What I am using to define mass, as is the common definition physicists use, is the invarient mass. Which is also equal to the "rest mass" of a system.

Does that help some?
In Newton physics, momentum p= mv

In SR momentum p = mvy

where y is 1/ Sqrt( 1 - v^2 / c^2)

Now that could be interpreted as p=(my)*v or p=m*(v*y)

Now if we measure v using the definition of distance over time and measure it to be v how do we justify that it not the mass that has increased?

Well we use a little trick where we measure distance using our rulers and time using a clock attached to the moving mass. Time as measured by the moving clock is time dilated so using this way of measuring velocity (our rulers, their clock) the velocity is greater by a factor of y. If the object was moving at the speed of light the time measured by the moving clock would be zero and by this definition of velocity the speed would be infinite. This slightly odd way of measuring velocity allows us to consider mass to be constant, which is convenient for the maths of calculating momentum etc. Using the concept of relativistic mass makes the calculations a little more difficult because mass parrallel to the motion is different to mass transverse to the motion. It is (in my opinion) a matter of interpretation and mathematical convenience. However, because most modern texts adhere to the convention that mass is constant and velocity and momentum are measured using the proper time of a co-moving clock(your rulers, their clock), it is best to get used to that interpretation, if you are going to be able to understand mainstream modern textbooks on the subject.

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Hi all - this (seeming) paradox has been brought up on a few other forums with less than illuminating results. I'm curious to see what the solution is.

Okay, so, you're watching a massive object approach the speed of light. From your reference frame, you are measuring the increase in apparent mass due to relativistic effects. This thing goes faster and faster and gains more and more energy, and so appears to get more and more massive. At some point, this thing will appear to be so massive that it should collapse under its own gravitation to form a black hole. However, from *its* reference frame, its apparent mass is constant - just its rest mass - and so clearly no black hole forms. That is, from one reference frame this thing collapses, while from the other reference frame it does not: paradox.

How is this reconciled? Where am I going wrong?
Each additional energy transfer takes longer as the speed of the object increases, producing slower rate of acceleration, which is interpreted as more inertial resistance.

Particle accelerators work with this condition. Try internet search on this subject.

Hi all - this (seeming) paradox has been brought up on a few other forums with less than illuminating results. I'm curious to see what the solution is.

Okay, so, you're watching a massive object approach the speed of light. From your reference frame, you are measuring the increase in apparent mass due to relativistic effects. This thing goes faster and faster and gains more and more energy, and so appears to get more and more massive. At some point, this thing will appear to be so massive that it should collapse under its own gravitation to form a black hole. However, from *its* reference frame, its apparent mass is constant - just its rest mass - and so clearly no black hole forms. That is, from one reference frame this thing collapses, while from the other reference frame it does not: paradox.

How is this reconciled? Where am I going wrong?
The answer is simple: the black hole forms through increased proper mass (i.e. the mass in the reference frame of the object).
Increased relativistic mass (i.e. the mass of the object as viewed from a moving frame wrt the object) does not contribute to forming a black hole. BTW, relativistic mass is an outdated term.

So, the correct answer is that there is no black hole forming in either frame, therefore, there is no paradox.

Hi all - thanks for the responses. It seems that the paradox has been reconciled (i.e. shown not be paradoxical) in a number of ways. However, because of my limited understanding of the subject area, I might need one more confirmation to tie it all together...

To restate, I see the problem as:
1. Energy gravitates.
2. The energy of an object can differ between reference frames.
3. Therefore, the gravitational strength can differ between reference frames.
4. Sufficient gravitational strength causes black hole formation.
5. Therfefore, one reference frame can see an object form a black hole, while another reference can see the same object not form a black hole.

In response to Dale, I recognize that length contraction would not form a black hole; my thinking was more along the lines of increased gravitation from the increased (kinetic) energy.

In response to 1effect et al., I've been under the impression that relativistic mass *does* gravitate since mass-energy in general gravitates [post #2 at physics forums dot com slash showthread.php?t=35884] (can't post the proper link because I'm under the 15 post minimum). I recognise the distinction between rest mass and relativistic mass (the moving object doesn't really have more "stuff"), but since the moving object has more energy, and energy gravitates...

Don't mean to belabor the point - just want to get this straight. Thanks again to all.

Hi all - thanks for the responses. It seems that the paradox has been reconciled (i.e. shown not be paradoxical) in a number of ways. However, because of my limited understanding of the subject area, I might need one more confirmation to tie it all together...

To restate, I see the problem as:
1. Energy gravitates.
2. The energy of an object can differ between reference frames.
3. Therefore, the gravitational strength can differ between reference frames.
4. Sufficient gravitational strength causes black hole formation.
5. Therfefore, one reference frame can see an object form a black hole, while another reference can see the same object not form a black hole.
I anticipated that you will go here. The attraction is a function of proper mass, so it does not increase with increased relative speed. I hope that this takes care of all your misconceptions.

I anticipated that you will go here. The attraction is a function of proper mass, so it does not increase with increased relative speed. I hope that this takes care of all your misconceptions.
thats not entirely true though. Personally, I have no idea why a black hole wouldnt be created (I read the FAQ thing on it here years ago before I took GR and SR) because energy DOES cause gravity. Photons create a gravitational field even though their proper mass is 0. Also, other types of non-massive energies can increase the gravitational mass (i.e. binding energies especially since the mass of composite particles like protons DOES NOT equal the sum of the masses of the quarks that compose it).

thats not entirely true though. Personally, I have no idea why a black hole wouldnt be created (I read the FAQ thing on it here years ago before I took GR and SR) because energy DOES cause gravity. Photons create a gravitational field even though their proper mass is 0. Also, other types of non-massive energies can increase the gravitational mass (i.e. binding energies especially since the mass of composite particles like protons DOES NOT equal the sum of the masses of the quarks that compose it).
You are not disproving my post, read it carefully and you will understand why. :-)

1. Energy gravitates.
2. The energy of an object can differ between reference frames.
3. Therefore, the gravitational strength can differ between reference frames.
4. Sufficient gravitational strength causes black hole formation.
5. Therfefore, one reference frame can see an object form a black hole, while another reference can see the same object not form a black hole.
Item 5 is incorrect.

Energy-momentum is related to gravitational strenth not just energy. Energy is not an invariant quantity unlike gravitational strength, e.g. spacetime curvature.

5. Therfefore, one reference frame can see an object form a black hole, while another reference can see the same object not form a black hole.
This is an incorrect conclusion. The laws of physics are the same in all reference frames. A black hole behaves differently to a non-blackhole and that would violate that very basic principle.

There are different types of energy and they do not all increase with relative motion. Enrgy associated with motion relative to the observer increases but internal energies, such as potential energy, binding energy and thermal energy can decrease.

One definition of energy is work also defined as force times distance. An expanding pistonn of a cylinder parallel to the axis of motion expands less by a factor of gamma showing that work done in the moving frame is less than the work measured in the proper frame. Force trnasverse to the motion is known to reduced by gamma while length remains constant tranverse to the relative motion axis. Again the thermal work done (energy) in the moving frame is less by gamma.

My previous post is the most intuitive explanation of why a black hole does not form purely due to relative motion. Can you show where that explanation is wrong?