The output of the full nonlinear Einstein equations from a metric perturbation h
Let us have a perturbation h of the flat Minkowski metric η. The Einstein tensor calculates a stress-energy tensor T from the metric
g = η + h:
Above, R is the Ricci scalar curvature, and we assume that the cosmological constant Λ is zero. The constant
κ = 8 π G / c⁴ = 2.08 * 10⁻⁴³ * 1 / N,
where G is the gravitational constant.
The stress-energy tensor T tells us what kind of a matter content in space would create the metric perturbation h to an otherwise empty Minkowski space.
The Einstein-Hilbert action for a spherical mass
The Einstein equations are derived from the Einstein-Hilbert action. For a lighweight spherical mass they produce the same field as the Coulomb field is for the analogous electric charge. Why is this?
In the action, the Ricci scalar R can be understood as the energy density of the gravity field. It is "potential energy". Adding Ricci curvature allows us to drop the mass-energy to a lower potential, i.e., we save in the "potential energy" of the mass-energy in the matter lagrangian LM. The price that we pay is that the potential energy in R grows.
We can probably derive the Coulomb field using a similar procedure. If we have a "bare" charge Q, we can drop it to a lower electric potential by creating an electric field around it. The price we pay is the energy that is required to create that electric field.
The energy of a wave in a rubber sheet model: "sound" waves transfer a lot of energy
Two years ago in this blog we calculated that the large energy density of a gravitational wave is due to the fact that it can stretch spatial distances.
Let us use a rubber sheet model of gravity. Let us have two mass systems A and B relatively close to each other. We try to transfer energy from A to B as efficiently as we can.
A B
● ●
● ●
rubber sheet
The systems A and B could be so close to each other that the energy transfer is mostly through (moving) static fields, and there is not much of a "wave" aspect.
In the rubber sheet model, we are able to transfer energy in waves which stretch the rubber sheet. Is this analogous to general relativity?
Maybe we can transfer much more energy in the "sound" waves (horizontal stretching) in the rubber sheet than in transverse waves (up and down movement of the sheet). This analogy would explain why the energy density of a gravitational wave is much larger than that of the analogous electromagnetic wave. Fundamentally, it is due to more degrees of freedom in the stretching rubber. Each degree of freedom opens a new "channel" to transfer energy.
The factor 16 might come from the 16 components of a tensor. If an electromagnetic wave uses only one channel to transfer energy, and a gravitational wave uses 16 channels, then the transfer speed could be 16-fold. A wave which stretches a rubber sheet in one direction involves many more distortions: there is shear involved, too.
Conservation of "energy" in the Einstein-Hilbert action
The action probably implies various conservation laws. One such law is conservation of "energy", where we mean an analogue of energy for an isolated system which sits in an asymptotic Minkowski space.
The ADM formalism proves that there actually is such a law, if we define the "energy" of a system S by the metric which it creates far away. The metric is close to a Schwarzschild metric for a certain mass M. The energy of the system can be defined as M c².
S₁ S₂
● ~~~~~~> ●
E
If we have a subsystems S₁ and S₂, and S₁ sends the energy E to S₂ through gravitational waves, then, obviously, the waves must generate far away the metric which corresponds to the mass-energy E. Otherwise, conservation of energy would be temporarily broken.
However, we are not happy about this reasoning because we must use the powerful result of the ADM formalism. We should find a simpler explanation.
Energy shipping in a Coulomb-like field and the Ricci curvature: an optical model of gravity
In this blog we have found grounds to claim that the apparent "metric" in the Schwarzschild solution is simply various inertia effects of the gravity field. The true underlying metric is Minkowski.
Let us concentrate on how the metric bends light. The bending is due to the local time running slowly close to a mass, and also due to spatial distances stretching.
We probably can deduce from that how the metric affects orbits of massive particles.
Hypothesis 1. Energy shipping in a force field whose potential is of the form ~ 1 / r, cannot "focus" or "defocus" light outside the charge. This implies that the Ricci curvature of the simulated "metric" is zero outside matter.
In gravity, we are mainly interested in the fields generated by mass-energy and pressure. We will forget about shear stresses for now.
The newtonian potential for a static mass M is ~ 1 / r. If we have a spherically symmetric mass M, then our hypothesis implies that the Ricci curvature is zero outside M.
The "potential" associated with pressure inside matter seems to be ~ 1 / r, too.
What about a mass M which is not symmetric? Maybe the "focusing" power can be "summed" from each mass element dM? Let us assume that the fields are weak.
Hypothesis 2. The "focusing" for light can be "summed" from each mass element dM of a larger mass distribution M, if the fields are weak.
Hypothesis 2 needs some tuning. We probably can sum the perturbations to the metric of time, but the perturbation of the spatial metric has to be deduced from energy shipping. Is there a reason why the Ricci curvature would still be zero in the surrounding empty space if we have two point masses?
Hypothesis 3. For a static mass distribution M, the Einstein field equations calculate the Ricci curvature right. They do it right both for the mass-energy density ρ and for the pressure p inside matter. The curvature is zero outside M, and the equations determine the correct non-zero value inside the mass distribution M.
Here we have an optical model for gravity. Each mass element dM creates a "lens", which makes rays of light to bend around dM just as described by the Schwarzschild metric. Summing the optics of these lenses is not straightforward, though.
__________
/___\___/___\
Above we have focusing lenses, which when summed, form a straight glass window. "Summing" of lenses is not simple.
The Einstein field equations "sum" the Ricci curvature caused by each mass element dM. Could this be the correct way to sum lenses?
focusing
/ \
/ ● \ dM
| |
| |
rays of light
In this case, we would concentrate on the focusing power of each mass element dM when rays of light pass it from different sides, and afterwards adjust the metric outside the mass distribution M in such a way that the Ricci curvature is zero there.
Conclusions
Let us close this blog post. We are not ready to study the 16-fold problem yet. First we have to figure out the exact relationship between our own inertia concept and the Ricci curvature of general relativity.
No comments:
Post a Comment