Re-Examining Black Holes and the Standard Model

[PAllen]

I have no problem in looking at events from different reference frames - and each one will yield a correct view of events.
So, in an instant, the rest mass is reduced by 98% - but the mass apparent to you in orbit does not change - and, until the poles move a substantial distance, there is little affect on your orbit. That's because in your reference frame, the total mass has not changed.

In fact from any inertial reference frame, the mass has not changed nor has there been a violation of the conservation of momentum - although not everyone will agree on the total mass or on which poll is more massive.

This is simply wrong. Rest mass in SR is a synonym for invariant mass. The invariant mass of the exploded Earth has not change at all from the explosion (and will never change if you include all the components). Different flyby observers will attribute different energy to the Earth (exploded or not) but will all agree on its invariant mass (pre or post explosion), and that this does not change (if you include all components of the system).

 

[PeterDonis]

in an instant, the rest mass is reduced by 98% - but the mass apparent to you in orbit does not change

Gravity does not depend on rest mass in GR; it depends on the stress-energy tensor. The "mass" of the planet, as measured by you when you measure your orbital parameters and apply Kepler's Third Law, didn't depend solely on its rest mass even before the explosion; even in the simplest case of a spherically symmetric perfect fluid, the fluid's pressure also contributes to its stress-energy tensor.

Also, how does any of this relate to your original scenario, which I responded to before?

 

[PeterDonis]

I have no problem in looking at events from different reference frames - and each one will yield a correct view of events.

Yes, but if you consider my description of your scenario in the rest frame of the gravitating object, you will see that it requires the description from your rest frame (in which the object is flying past you at near the speed of light) to be different from the one you gave in your previous post: the apparent "mass" of the object as measured by its gravitational effect on you does not increase without bound.

The more general rule, which I mentioned in my post a few minutes ago, is that an object's gravity is not determined by its relativistic mass (or by its rest mass); it's determined by the object's stress-energy tensor.

 

[PAllen]

Yes, but if you consider my description of your scenario in the rest frame of the gravitating object, you will see that it requires the description from your rest frame (in which the object is flying past you at near the speed of light) to be different from the one you gave in your previous post: the apparent "mass" of the object as measured by its gravitational effect on you does not increase without bound.

The more general rule, which I mentioned in my post a few minutes ago, is that an object's gravity is not determined by its relativistic mass (or by its rest mass); it's determined by the object's stress-energy tensor.

Hmm. The oft referenced paper:

https://home.comcast.net/~peter.m.brown/ref/mass_articles/Olson_Guarino_1985.pdf

derives that the active gravitational mass of flyby object on an initially stationary test body is mγ(1+β²). The ultra-relativistic limit is 2γm. This says it does increase without limit.

 

[PAllen]

Hmm. The oft referenced paper:

https://home.comcast.net/~peter.m.brown/ref/mass_articles/Olson_Guarino_1985.pdf

derives that the active gravitational mass of flyby object on an initially stationary test body is mγ(1+β²). The ultra-relativistic limit is 2γm. This says it does increase without limit.

Here is an explanation that squares this result with Peter's argument in #30. I will refer to body (for a massive body) and particle (for test body). I will assume that the speed of flyby is ultra-relativistic. Then, in the frame of the body, you have a deflection that 2 times Newtonian expectation. In the frame of the particle, since the deflection is primarily orthogonal to the relative motion, you have the same amount of deflection occurring in a time interval shorter by a factor of γ than observed in the body frame. Thus, the active gravitational mass (in particle frame), by this definition, of the body is increased by γ over the body frame; thus 2γ times the Newtonian expectation.

 

[.Scott]

...So, in an instant, the rest mass is reduced by 98% - but the mass apparent to you in orbit does not change - and, until the poles move a substantial distance, there is little affect on your orbit. That's because in your reference frame, the total mass has not changed.

In fact from any inertial reference frame, the mass has not changed nor has there been a violation of the conservation of momentum - although not everyone will agree on the total mass or on which poll is more massive.

This is simply wrong. Rest mass in SR is a synonym for invariant mass. The invariant mass of the exploded Earth has not change at all from the explosion (and will never change if you include all the components). Different flyby observers will attribute different energy to the Earth (exploded or not) but will all agree on its invariant mass (pre or post explosion), and that this does not change (if you include all components of the system).

First of all: Oops, you blew up the Earth. I was hoping for some for some other planet-size bomb.

For simplicity, I will change things slightly. Instead of an orbiting observer, I will have the observer at zero velocity relative to the Earth when the explosion occurs.

But using Earth, here are the specifics of the explosion:
Before the Explosion:
Rest Mass of Earth: $6\times10^{24}Kg$
Velocity of Earth relative to our observer: 0
After Explosion:
Rest Mass of each remaining piece: $6\times10^{22}Kg$
Velocity of each polar fragment relative to our observer: 0.9998c
Total Rest Mass: $12\times10^{22}Kg$
Total Mass relative to observer: $6\times10^{24}Kg$

Unless you disagree with these calculations, note that the rest mass changed dramatically during the explosion. But the relativistic mass remained constant.

 

[PeterDonis]

In the frame of the particle, since the deflection is primarily orthogonal to the relative motion, you have the same amount of deflection occurring in a time interval shorter by a factor of γ than observed in the body frame.

Yes, I agree with this. To me, it just points out another problem with trying to apply intuitions about "gravitational mass" in relativistic scenarios: the amount of deflection remains finite even though the "active gravitational mass", if it's defined as "deflection per unit proper time", increases without bound.

 

[PAllen]

First of all: Oops, you blew up the Earth. I was hoping for some for some other planet-size bomb.

For simplicity, I will change things slightly. Instead of an orbiting observer, I will have the observer at zero velocity relative to the Earth when the explosion occurs.

But using Earth, here are the specifics of the explosion:
Before the Explosion:
Rest Mass of Earth: $6\times10^{24}Kg$
Velocity of Earth relative to our observer: 0
After Explosion:
Rest Mass of each remaining piece: $6\times10^{22}Kg$
Velocity of each polar fragment relative to our observer: 0.9998c
Total Rest Mass: $12\times10^{22}Kg$
Total Mass relative to observer: $6\times10^{24}Kg$

Unless you disagree with these calculations, note that the rest mass changed dramatically during the explosion. But the relativistic mass remained constant.

Of course I disagree because you are making the same mistake again. The invariant mass of a system is not the sum of rest masses of components. It is the norm of the sum of 4-momenta. The sum of rest masses of multiple bodies in relative motion is a quantity of no significance in SR. Since momentum is conserved, the total momentum of all explosion products is zero as it was before (they are going off in all different directions, at magically the same speed, but we can accept this conceit). This means the total 4-momentum is (<total energy>,0,0,0) with time component first. The norm is then <total energy>/c^2. Thus, the invariant mass has not changed (and is invariant).

Since this is a really basic mistake, I am curious if you have ever formally studied SR.

 

[PeterDonis]

the rest mass changed dramatically during the explosion

The rest mass of individual pieces changed, yes. But the rest mass of the total system did not; that was PAllen's point. Rest mass is not additive: a system composed of multiple pieces in relative motion can have a rest mass (a better term is "invariant mass", as PAllen said) that is not the sum of the rest masses of the individual pieces.

 

[PeterDonis]

The OP's question has been answered and he hasn't been back in a while, so this thread is closed.