The Circle and the Sphere

Consider, for a moment, half a sphere, as a contemplation of gravity; a bowl shape.

Consider the circle, whose radius r is the distance from the center of the gravity well.

How do you measure r?  Do you measure from the bottom of the bowl, following the curve of the walls of the bowl?  Or do you measure from what would be the center of the whole sphere?

How do you measure the surface area of the circle?  Do you measure the curved space of the bowl?  Or the space of an imaginary flat disc?

These questions are important, and as far as I can tell, don't get a lot of attention.  If you measure the radius from the bottom of the bowl, following the curve of the bowl - and if you measure the surface area as the curved space - then you get a fascinating and important result, in terms of understanding what gravity is, and what it does.

I have read arguments from physicists that gravity is not self-interfering.  But there is an important phenomenon, which in view of the circle, casts doubt on that statement: Negative mass binding.  The gravity exerted by a body, at a given distance, is less than twice the gravity that would be felt if you halved the mass of the body.

But it's easily explained if you think in terms of our circle.  First, the distance is greater - follow the curve of the sphere, rather than moving through imaginary space.  But using our increased distance, the surface area of the circle does not increase as pi*r^2 - or, considering a sphere in three dimensional space, the volume is not actually 4/3 pi 4^3, because in either case, the volume under consideration is not in fact Euclidean.  The greater the gravity - the greater the curvature - the less accurately these equations describe surface area, or volume, respectively.

At some distance, however, these equations start being more accurate.  They're more accurate still if you remove the excess surface area, or volume, respectively.  You can, in a sense, segregate out the excess volume, and consider only the flat-space amount of volume.

Now, if you consider curvature as mass-energy over unit volume - that is, if you think of gravity in terms of curvature, and curvature to be associated with a specific amount of mass-energy, this implies a neat little trick: If you can successfully calculate the amount of excess volume, and the amount of mass-energy associated with that volume, you can simply subtract out this amount from the mass-energy.

Which is what we do.  Gravitational binding energy; negative mass binding.

Now, what's interesting is that we have a much simpler way of calculating the same thing; we just figure out how much energy is necessary to take everything apart again.  Notice the implication here; compressing a volume of space releases energy.

Consider mass, whose very existence represents - through gravity - a compression of a volume of space.

There's an important contradiction there.  But let's set it aside for the moment.  Two questions, for those who are feeling iffy about all of this: When you measure distance from a large mass, do you measure the distance across curvature, or the distance across flat space?  Why should gravity traverse an imaginary flat space, rather than the curved space?

Now, thinking of these circles in terms of other things may also be illuminating.  Or confusing.  Consider the circumference of our circle; is an event horizon the point at which the circumference does not vary with distance-over-curved-space?  Does this imply infinite volume?  Does it imply infinite distance?  Can you have infinite distance but finite volume?  What about the case when circumference increases as distance decreases?

In order: Not necessarily, not necessarily, yes, and "This is a very important case to consider."

The point at which circumference no longer varies is the point at which time and distance are no longer distinct dimensions; they are the same dimension.  To a significant extent, at this point, it's no longer really meaningful to talk about volume; from one perspective, there's finite volume, and from another perspective, there's infinite volume.

An event horizon, once you consider it in terms of our circle, is just a cylinder whose length is at a tangent to our three dimensions.  The infinity arises from trying to measure the cylinder from the perspective of the dimensions is it tangential to.  From the perspective of the cylinder itself, it has a finite volume.

That said, the locally-measured distance and time it takes for a particle to traverse the cylinder can still be infinite, even though the cylinder has a finite extent and volume.  That, however, is a contentious statement, and most physicists are going to disagree with it.  We'll move on for lack of a useful metaphor I can use to describe the notion of time here; I can only gesture at tan(x), which describes an infinity where another perspective sees a finite space, and say "Look here, and realize that time and distance have a similar relationship."

Returning to the contradiction, the problem arises in that mass is a contraction of space, and yet contracting space reduces, rather than increases, the amount of mass involved, from a certain perspective.  More mass = less mass.  Mass is its own opposite.

"Hang on," you start to say, "that isn't what that means at all.  Even accepting the idea that negative energy binding can be described as a variation in distance, mass isn't it's own opposite; rather, some mass is converted to energy.  Gravity is caused by mass and energy, and you're ignoring the energy that escaped the system - the binding energy - when the masses coalesced."

Which is all well and good, except that the change in mass, once you view it in the distance framework, is all imaginary; all the mass is still there.  The lost mass had to have been energy - and the energy that was released, was the potential energy of the system.  Potential relative to what - mass relative to what.  Well, if we consider an object formed from two objects that fell into one another, the potential relative to one another.

From an outside perspective, treating the two falling objects as a black box, energy is released, so the mass goes down.  Simple, right?

Well, from the distance-perspective, what is the potential energy?  What is the energy that is released?

Hopefully you can start to see the shape of where this is going: Energy can be expressed purely in terms of changes in distances between things.  Energy doesn't "add" to the curvature of space-time through a secondary process; energy -is- curvature in space-time.  A wave is just a moving change in distance.

And, realizing that energy and mass are the same thing, this leads to another conclusion: Mass doesn't curve space-time.  Mass is space-time curvature.  Mass doesn't cause gravity, mass is gravity.  And in that perspective, the curvature in question isn't missing - it's someplace else.  Yes, the energy left, but, if we count only cubic meters, every cubic meter is still accounted for; what has changed is where each cubic meter is.

Well, kind of.