General relativity, will we learn otherwise?

[Majik]

Originally Posted by HydrogenBond

If you look at the vector addition of gravity in a planetary sphere, the vectors cancel in the center to give zero gravity, based on Newtonian gravity. One thing nobody talks about is where is the energy going?

There is also a vector cancelling of gravity out in empty space at Lagrange points between massive bodies. For example, we park satelites at a particular distance between the sun and the earth because the gravitational pull of the sun is balanced by the earth, and it takes less fuel to maintain its orbit there. Where does the potential energy go at those points?

[CraigD]

Originally Posted by HydrogenBond

If you look at the vector addition of gravity in a planetary sphere, the vectors cancel in the center to give zero gravity, based on Newtonian gravity. One thing nobody talks about is where is the energy going? In other words, if we took apart a planet, each piece of matter could generate gravity.

The energy – or, specifically, the gravitational potential () which give the difference in potential energy due to gravitational force for a test body of unit mass for any pair of points – is precisely what the preceding approximations have addressed, by doing literally what HBond describes – taking a planet – or rather, the Sun – apart into several thousand evenly-spaced pieces.

The gravitational potential energy doesn’t go anywhere mysterious. The greater the distance of a test body from the center of mass of a large body (Sol in the previous examples), the greater its potential energy, which corresponds to a smaller value of U. U reaches its maximum at distance zero, while the net acceleration due to gravity reaches zero there.

A potential source of confusion is the sign of U, which for gravitational time dilation calculations is positive, while for classical Newtonian calculations, is negative. So, using the values from post #39 a 1 kg body 90 solar radii from the center of Sol has about 2,118,027,173 - 1,906,223,767 = 211,803,406 Joules less, not more, gravitational potential energy than a 1 kg body at 100 solar radii. Intuitively, the “higher” something is, the greater its GPE. A body at a point with U=0 has the highest GPE it can have.

To better illustrate this I’ve added a column for net acceleration to the table in post #39, and changed its scale to make it more readable. It still suffers from small calculator lack of precision and “granularity” relics, resulting in its gravitational potential actually maxing at 0.1 solar radii from center, where the lattice of pieces align, rather than at 0, but it’s less than a 1% inaccuracy.

Originally Posted by HydrogenBond

If place them all in a sphere we lose gravity in the center. What this implies is potential energy is being lost.

No, it does not imply that.

Energy is NOT equivalent to force, but to the sum of force times distance over which the force is applied.

Assuming there are no other bodies in the universe, a body placed at and at rest relative to the center of mass of a second body has lost all of its gravitational potential energy. The energy has gone into whatever work was performed to place it there. Were no work done placing it there, or at any point less distant than some starting point – that is, has the body free-fallen there - the lost potential energy would be in the form of relativistic kinetic energy, which for low speeds can be accurately described by the classical formula , or for any speed, an increase in the moved body’s mass given by .

Here’s an example illustrating the difference between force and work/energy: I push a car up a hill, giving it potential energy. The top of the hill is level, so the car experience no net force in its allowed direction of movement. If I release the car, it won’t do any work (roll down the hill). However, it still has the potential energy I gave it – if I push it over the edge of the hill, that potential energy, plus any I added getting it rolling, minus losses do to friction, will be transformed into kinetic energy.

[modest]

Originally Posted by CraigD

Still, the gravitational time dilation in Sol, or even the most massive stars, remains very small, far below what I think could be detected by observation.

I agree. Unless you're modeling the collapse of a supermassive star, this is not a useful or significant variable in discussing the rate of fusion.

-modest

[HydrogenBond]

Let me give an analogy for the virtual GR affect of Newtonian gravity. Say we had a generator with a single rope loop around a pulley. As you pull, the gear turns and the rope recycles within the loop. If we pull equally from opposite sides of the loop, i.e., rope top and rope bottom, there is no net force being applied to the generator, due to force vector addition. Yet we could generate energy constantly, using zero net force. The center of gravity, is zero net force, but it powers the GR generator.

The might be explainable with a particle wave example. If gravity force was only expressed with particles, hypothetically, the center would be traffic central since it has the most combinations of particle interaction at the shortest distance. If we use the wave nature, only, the waves cancel in the center. Here is the paradox, traffic central should have tons of particles but the waves say there should be no particles. To maintain consistency, the particles can not be there. But over time, these particles are constantly being exchanged while constantly ending up in wave nothingness. Since the particles reflects force and distance or energy, this energy needs to appear elsewhere, to be conserved. Since it can not be expressed within the wave nothingness, it comes out as GR, as different type of wave.

[modest]

Originally Posted by CraigD

(As I don’t know a formula for the gravitational potential inside a sphere of uniform or other density, I’ve fallen back to a numeric approximation, representing the sphere as a cloud of 26,252 point masses.

derived on Gravitational potential due to rigid body

gives equivalent answers (to yours). It assumes uniform density.

-modest