Can the fabric of space-time be contoured into hills instead of just wells?

Question

Einstein's general theory of relativity states that gravity is the distortion of space-time into gravity wells. In order to illustrate this, a flat plane is used to represent undistorted space-time grid. Around mass the flat space-time fabric is shown sinking down into a parabolic well which encloses the mass.

According to my primitive understanding of this representation:

1. The depth of the well is proportional to the total mass of the object.

2. The diameter of the well at the top is proportional to the volume of the mass.

Thus a post single star mortem black hole's space-time distortion would be represented by a narrow but very deep parabolic well depression in in the otherwise flat plane grid fabric.

A star like our sun's gravity well would be illustrated by a wider diameter at the top of the well but not as deep as the above mentioned black hole's well.

On the other hand, an entire galaxy gravity well would be illustrated by an extremely wide but shallower parabolic gravity well depression.

Please indicate if my understanding of the space-time fabric distortion above is in error.

I was wondering if there was more to this illustration that can help explain dark matter and dark energy.

Dark matter was theorized to explain the intra galactic phenomenon of orbital speed of stars in the outer arms of the galaxy being greater than the escape velocity for the galaxy at the said distance from galaxy center. The galaxy mass must be much greater than the observed mass to keep these orbiting stars from flying off due to centripetal force.

Dark energy was theorized in the attempt to explain the observed deep space red shift data that the expansion of the universe is accelerating. Sort of like an anti-gravitation force pushing the distant galaxies apart.

My intuition tells me that this illustration is over-simplified and may be omitting a more comprehensive overall effect on how mass distorts space-time. Indeed, completely flat planes are unnatural as are parabolic depressions without rims along the perimeter.

Consider that just as mass causes wells and depressions in the space-time fabric, why can't lack of mass in intergalactic space cause a hill or bulge in the other direction? Such a bulge or hill would have the opposite effect of mutual attraction of masses but rather mutual repulsion of them. The bulge or hill in space-time would be significant only at extremely long distances between galaxies where the inter-galactic space mass density would be extremely low to almost perfect vacuum. This mutual repulsion of distant galaxies due to the inflation of the space-time grid would explain the accelerating expansion of the universe without resorting to dark energy.

Accordingly, if we can imagine a space-time inflationary hill along the perimeter of a galactic gravity well, the extra relative depth from the top of the hill to the space time grid plane would explain the greater overall mutual attraction throughout the galaxy. The gravity well would be deeper due to the hill on the perimeter, thus raising the escape velocity for masses in the galaxy, eliminating the need to explain with dark matter.

Extending this train of thought using nature as example: Just as a mass dropped in a pond of still water causes concentric ripples across the planar water surface, a galaxy would also cause a multitude of concentric hills (inflation) and valleys (compression) of the space time grid. This would start with the the largest inflationary bulge at the rim of the gravity well of the galaxy and concentric alternating valleys and hills of decreasing size with increasing distance from the galaxy.

The distance between the peaks and valleys of space time would be astronomical, thousands if not millions or billions of light years.

We can also imagine at some point in inter-galactic space the ripples merging into huge hills in space time grid causing the accelerating expansion of universe noted above. The large valleys are where we would find galaxy clusters.

Is there any reason why the Einstein model of gravity is only the case of the compression of space-time in our realm of experience?

Answer 1

If you take an embedded spacetime slice and turn it upside down, it has the same intrinsic geometry, and therefore the same gravitational effect. A "hill" is the same as a "well."

The important property of the well/hill shape is that any circle around the center has a circumference smaller than π times its diameter. If the mass in the center was negative, then (theoretically, plugging in a negative value to the Schwarzschild solution) the circumference would be larger than π times the diameter. I don't know whether you'd be able to embed this exactly, but I imagine it would look something like a crocheted hyperbolic plane. It wouldn't look like a hill or well.

A locally higher value of the Newtonian gravitational potential does have an antigravity effect, a graph of it does look like a hill, and if you imagine the graph as a rubber sheet you get a surprisingly accurate model of Newtonian gravity. But spacetime in general relativity doesn't work that way.

Answer 2

I personally hate that rubber sheet analogy, guess what pulls on stuff to keep them on the sheet? Gravity! A circular argument, using gravity to model gravity, but anyways...

There is a flaw in your model, if you wanted to run with it as far you have, what you would do is find the Schwarzschild radius of whatever mass you were modelling, then project your well from that. Then for the portion where your well was narrower than the object your well would go up rather than down, ending up back at zero in the very centre. Thus the width of your well would depend ONLY on the mass and not the volume.

Answer 3

Time dilation is not really visualized in the hill/well analogy. So, some other counter argument to the proposal should be used to argue against gravity hills. In fact a gravity hill could be a region of space where time passes faster.