Minkowski-minkowski thin shell paradoxon?

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In summary, the spherically symmetric infinitesimally thin shells can be described via the well known junction formalism of Israel. The equation for the motion in such a shell is the first component of the Einstein equation, in the thin shell limit. For a shell with Schwarzschild mass mc+mg, there is only one solution for r, mr, and mg. If we assume the dust case (when all mass parameters are constant during the motion), then x=y implies z=0. However, if we assume Schwarzschild-Schwarzschild junction, then mg can be negative and the positive energy theorem is still valid. This means that on both horizons, a normal situation exists.
  • #1
mersecske
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Spherically symmetric infinitesimally thin shells
can be described via the well known junction formalism of Israel.
Let us consider such shell in vacuum,
this means that on both sides of the shell we have Schwarzschild spacetimes.
One of the dynamical equations is the first component of the Einstein equation
in the thin shell limit:

sqrt(1-2mc/r+v^2) - sqrt(1-2(mc+mg)/r+v^2) = mr/r

where r is the circumferential radius;
v = dr/dtau, and tau is the proper time of the shell;
mc is the central Schwarzschild mass parameter;
mg is the gravitational mass of the shell, this means
that the outer Schwarzschild mass parameter is mc+mg;
and mr is the rest mass of the shell, mr > 0;

Let assume the dust case,
when all mass parameters are contant during the motion.

In the case of Minkowski-Minkowski junction mc=0, mg=0
and the equation reduces to

sqrt(1+v^2) - sqrt(1+v^2) = mr/r

therefore mr=0 is the only solution.
This is what we expect becaue the whole space is Minkowski.

But let us consider Minkowski-Schwarzschild junction.
In this case mc=0 but mg is not restricted.
The equation is

sqrt(1+v^2) - sqrt(1-2mg/r+v^2) = mr/r

The solution of this equation for mg is:

mg = mr(2r*sqrt(1+v^2)-mr)/r/2

We can see that for mr=0 we get mg=0,
but it is possible to set mg=0 with positive mr also!
For example with the initial condition v=0,
and mr=2r we get mg=0.
And since mass parameters are constant during the motion,
we get a moving shell solution with positive rest mass
but zero gravitational mass,
which means that the spacetime is Minkowski both inside and outside!
It is very strange if this is the reality.
We have energy in the spacetime
but the spacetime is Minkowski except on a singular hypesurface.
 
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  • #2
Suppose

[tex]\sqrt{1 + x} - \sqrt{1 + y} = z.[/tex]

Then, [itex]x = y[/itex] implies [itex]z = 0[/itex].

Rearrangement and squaring, however, gives

[tex]
\begin{equation*}
\begin{split}
1 + y &= \left(\sqrt{1 + x} -z \right)^2 \\
&= 1 +x -2z\sqrt{1 + x} + z^2 \\
y &= x + z\left(z - 2\sqrt{1 + x}\right) .
\end{equation*}
\end{split}
[/tex]

Consequently, [itex]x = y[/itex] implies [itex]z = 0[/itex] or [itex]z = 2\sqrt{1 + x}[/itex]. What happened?
 
  • #3
Thank you, sorry about this triviality
 
  • #4
But I have another question.
Now we assume Schwarzschild-Schwarzschild shell.
In this case mg can be negative,
and the positive energy theorem is not violated,
since mc + mg > 0 and mr > 0,
where mc is the central Schwarzschild mass parameter,
mr is the rest mass of the shell,
and mg is the gravitational mass of the shell.
How can we interpret this situation?
Above both horizon, this is a normal situation?
 

What is the Minkowski-minkowski thin shell paradoxon?

The Minkowski-minkowski thin shell paradoxon is a thought experiment in physics that explores the concept of time dilation and the principles of special relativity. It involves a hypothetical scenario where two observers, one on the inside and one on the outside of a moving hollow sphere, experience different perceptions of time due to their relative velocities.

How does the Minkowski-minkowski thin shell paradoxon challenge our understanding of time?

This paradoxon challenges our understanding of time by showing that time is not absolute, but rather relative to the observer's frame of reference. The concept of time dilation, where time appears to pass at different rates for different observers, is a key component of the paradoxon and can be difficult to grasp intuitively.

What is the significance of the Minkowski-minkowski thin shell paradoxon in the field of physics?

The Minkowski-minkowski thin shell paradoxon is significant because it helps us to better understand the principles of special relativity, which have been confirmed by numerous experiments and play a crucial role in modern physics. It also highlights the counterintuitive nature of these principles and the need to approach them with an open mind.

Is the Minkowski-minkowski thin shell paradoxon a real phenomenon or just a thought experiment?

The Minkowski-minkowski thin shell paradoxon is a thought experiment and does not have any empirical evidence to support it. However, the principles and concepts explored in the paradoxon have been experimentally verified, making it a valuable tool for understanding and furthering our knowledge of physics.

Are there any proposed solutions to the Minkowski-minkowski thin shell paradoxon?

There have been various proposed solutions to the Minkowski-minkowski thin shell paradoxon, including the idea that the observers inside and outside the sphere would not experience a paradox due to the effects of gravity. Another proposed solution involves considering the observer's relative accelerations, rather than just their velocities. However, the paradoxon remains a topic of debate and continues to challenge our understanding of time and relativity.

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