A paradox connected to General Relativity

[puppypower]

I am posting this paradox as a brain teaser

If we drilled a tunnel through the earth, to the other side, and measured gravity in the tunnel, gravity would be zero in the center of mass. This is Newtonian gravity and is connected to the vector addition of the gravitational force; cancels in all directions in the center.

That being said, when we plot the space-time well, associated with a stars and planet, why is the center of the star portrayed as the place of maximum space-time contraction, when the gravity is zero in the center? If we apply this to a black hole, its center of gravity also has zero gravity and its maximum gravity should be on the surface. Yet, we plot the space-time well as though the center is highest in gravity, even though it is zero.

Although gravity does not add up with respect to the direction convention of the space-time well, the well is consistent with the pressure gradient. The pressure is highest in the center and lowest at the surface. Why does the space-time well reflect pressure better than gravity?

 

[Ibix]

You appear to be mixing up ideas like time dilation that depend on gravitational potential with things like "gravitational acceleration" that depend on the derivative of the gravitational potential. See a recent thread in this forum started by DaveC426913.

Also black holes don't really have a well-defined centre inside the horizon - the singularity lies in the future, not the centre. So attempting to reason by analogy with "normal" astronomical bodies won't help.

 

[Dale]

The plot you are thinking of, when it is done correctly, represents the gravitational potential, not the gravitational field. The potential is indeed lowest in the center.

The gravitational force is the gradient of the potential, so it is zero in the center.

 

[Orodruin]

The plot you are thinking of, when it is done correctly, represents the gravitational potential

Which, it should be stressed, does not represent spacetime curvature. This

why is the center of the star portrayed as the place of maximum space-time contraction

is essentially just word sallad. ”Spacetime contraction” is not a standard term.

 

[PeterDonis]

If we apply this to a black hole

You can't. A black hole is not like an ordinary object. It doesn't have a spatial "center", and spacetime inside the horizon is nothing like spacetime inside an ordinary object.

 

[PeterDonis]

gravity does not add up with respect to the direction convention of the space-time well

What do you mean by this?

 

[sweet springs]

Although gravity does not add up with respect to the direction convention of the space-time well, the well is consistent with the pressure gradient. The pressure is highest in the center and lowest at the surface. Why does the space-time well reflect pressure better than gravity?

Let us climb on a mountain of conical shape. The top is flat and has lower air pressure than on the plane. I do not suspect low air pressure would effect space-time structure or metric at the summit, though you may do. We do not need weather reports saying Low or High pressure airs are coming to adjust clocks.

 

[PeterDonis]

I do not suspect low air pressure would effect space-time structure or metric at the summit

Why not? Pressure is part of the stress-energy tensor.

It is true that, here on Earth, the stress-energy effect of the pressure of air is negligible. But that's different from saying you "suspect" it wouldn't affect the curvature of spacetime at all.

 

[sweet springs]

Air pressure with Einstein's constant $$\kappa$$ could affect pace of time as you taught. We should bring clocks set in vacuum tubes to the summit in order to avoid pressure effect. Thanks.

 

[PeterDonis]

We should bring clocks set in vacuum tubes to the summit in order to avoid pressure effect

Apparently you didn't read the part where I said that here on Earth the effect is negligible. It's far too small to measure even with our most accurate clocks.

 

[sweet springs]

Ok, I will worry about air pressure effect in future when we have advanced technology to sense such a subtle difference.