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Author Topic: Problem with Birkhoffs theorem and the inside of a collapsing sphere  (Read 2393 times)

Offline Frodeborli

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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.

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.


 

Offline imatfaal

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Interesting.  my concern would be the extremes of the gedanken.  as other have pointed out in similar threads; do the impossibilities of the initial conditions render any insight as invalid?

your argument seems, to my untutored eye, to be sound.  from quick research the agreement of birkhoffs theorem with newtons shell theorem is a corollary of the main theorem so any challenging of that theorem would have to show the link (between theory and consequence) still holds for a collapsing sphere. 

this is an area that has great mathematical agreement - and it would seem to me that rigorous mathematical argument is a necessary part of any challenge. 
Matthew
 

Offline Murchie85

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Just to clarify,

Would this universe look something like my little doodle? In that the red stars represent the main 5, and the yellow stars are the distrabution? Also would the NESW stars be at the edge?

 

Offline Frodeborli

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Imagine a coconut. When you slice it in half, the white would be the outer stars, and where the coconut milk was is the location of the inner five stars. But yes, it would look similar to your illustration. The blue circle would represent the outer stars, the yellow stars would be distant stars in the Z direction.
 

Offline Frodeborli

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Interesting.  my concern would be the extremes of the gedanken.  as other have pointed out in similar threads; do the impossibilities of the initial conditions render any insight as invalid?

your argument seems, to my untutored eye, to be sound.  from quick research the agreement of birkhoffs theorem with newtons shell theorem is a corollary of the main theorem so any challenging of that theorem would have to show the link (between theory and consequence) still holds for a collapsing sphere. 

this is an area that has great mathematical agreement - and it would seem to me that rigorous mathematical argument is a necessary part of any challenge. 
Matthew

Notice that the scales I'm talking about here are immense. The distance between one edge of the sphere and the other edge would be billions of light years.

In more "real" scales, linear frame dragging of individual particles and relativistic mass can probably be ignored. In fact, many (most?) physicists ignore these effects because they are a source of errors in the formulas - and removing them does not change the end result in the physics we need when viewing orbital paths of galaxies, stars and planets. ( newbielink:http://en.wikipedia.org/wiki/Frame-dragging [nonactive])

If observed from a very large distance exterior to the sphere, these effect can probably also be ignored. I'm talking of small "window of time" which occurs when particles accelerate at different distances simultaneously. Both particles gain momentum, which in turn create increasing curvature of space. This increase in curvature propagates by the speed of light.

If two particles at two different distances from you are accelerating, the increase in momentum and frame-drag from the particle closest to you will affect you before the particle further away from you. A sphere is made up of particles.

I've not got the mathematical skills to prove this right now, and I am hoping to find somebody that will test it mathematically with me.
 

Offline LeeE

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Imagine an empty universe, where nothing exist and time stands still.

I'm afraid that any theory based upon the movement of time i.e. 'standing still' is fundamentally flawed.  Time, like space, does not move; it's just a direction in which you might move.  Indeed, if your universe is to be dynamic...

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Then add lots of stars

...you have to move along the time axis, for without doing so you must simultaneously have both an empty universe with no stars and at the same time, one that it is filled with many stars.  As soon as you write "Then" you have invoked movement in time.
 

Offline Frodeborli

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It seem that people are unwilling to imagine things here. I should never have mentioned time, as you keep telling me banal information like that the final configuration I was building could not happen? A perfect sphere can not exist in nature either, but scientists like Birkhoff spent years calculating on them. How can you know that about time, by the way? If everything moved at the speed of light since second 0, time would not have passed for you as an observer when all objects were in place. Not even gravity would have propagated to affect any of the other objects in the system. So can we please come back to the original point of my post?
 

Offline Murchie85

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Frodeborli,

Before I can comment on the general characteristics of your idea, first i would like to understand  why the 4 stars are moving away from each other while the rest of the stars are moving inwards? Wouldn't these 4 stars follow the same path as the rest of the sphere stars?
 

Offline PhysBang

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For the purposes of a thought experiment, I have no problem with the stopping of time. The problems start here:
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.
This is not relativity. General relativity was specially formulated so that the propagation of gravity is limited to the speed of light.
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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.
You are invoking a scenario that is not the ideal scenario of Birkoff's Theorem. Why should you be surprised when there are small differences?
 

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