Imagine an empty universe, where nothing exist and time stands still. Then add lots of stars of equal size, distributed in a symmetry around a spot that we call the center of our universe. Since time has not passed, no curvature (gravity) has propagated to affect any of the other stars. No outside force affect this universe, not even you as an observer.(adsbygoogle = window.adsbygoogle || []).push({});

Inside this hollow sphere of evenly distributed stars, we add five stars. One at the Center (C), and for simplicity we call the locations of the other four stars North (N), South (S), East (E) and West (W).

Remember that no time is passing yet, as we are still building our imaginary universe. No curvature of space have propagated from each of the stars, so nothing affects any of our stars yet.

Then we start time, simultaneously, for all stars in the system. What should happen is:

1. Light and gravity should propagate away from each of the stars. After some time, light from each of the stars should reach its closest neighbor. At the same time as light, curvature of space-time have propagated.

Each of the stars are now only affected by their closest neighbors gravity/curvature of space-time. This should trigger a slow collapse of our sphere. As more stars start affecting each other, the collapse should accelerate.

2. At a certain time, gravity from stars making up the sphere in the north region should reach our (N)-star. At the same time, the same happens for the (S), (E) and (W)-stars. The sphere should collapse slightly faster in these directions, but our four stars should begin accelerating toward the sphere-stars.

Now, all our five central stars are moving away from each other, while all stars making up the sphere is moving toward each other.

3. At one time, all stars in our universe will have extended their gravity field so that it affects all other stars in our universe. For the (N)-star, the gravity pull from all stars in the (N)-region equals the gravity pull from all stars in the (E), (S) and (W) region combined. This is Newtons theorem in essence. The gravity cancels out.

For a short time, in step 2 above, all stars inside the universe was moving away from each other. Once curvature of space-time from each star have propagated troughout the universe, all stars are affected by it equally in all directions - making the curvature of space-time flat. There will be no gravitational effect inside the sphere - except gravity from the five stars inside the sphere on each other.

Once the curvature of space-time have propagated, all gravity is instant. When a star moves, the direction of the gravity is immediately reflected for all other observers. Infinitely faster than light. The logic behind this is : once space is curved, everything is affected by the curvature. If one star is defined as stationary, and another star is moving toward it - both stars are affected by the change in curvature - because we can at any time redefine which of the two stars are stationary. This is relativity.

Birkhoffs theorem and Newtons theorem say that inside a hollow symmetric sphere, gravity is zero - and the above logic supports that. But I still have a problem with it.

All moving objects have momentum/energy. Energy contribute to curvature. All stars in the sphere have equal speed - so the effect should be the same - zero gravity. Birkhoffs theorem is still true. BUT:

As all stars in the sphere accelerate, so their momentum/energy increase. Increasing momentum/energy propagate at the speed of light. My intuition tell me that the increase in momentum should affect the closest stars inside the sphere first, and as long as the collapse of the sphere is accelerating - all stars inside should experience an outbound gravitational effect. Visually, this should lead to a redshift effect when watching other stars inside the sphere.

Where am I wrong in this last conclusion, that a sphere that is collapsing by its own gravity will NOT have zero gravity inside - contrary to Birkhoffs theorem.

Newtons theorem does not apply, as gravity propagates instantaneously by Newtons theories.

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# Problem with Birkhoffs theorem on the inside of a sphere collapsing by its own weight

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