The metric is supposed to determine the orbit of a freely falling particle of infinitesimal mass.
Extra inertia which is caused by pressure in a spherically symmetric mass: does the mass have time to adjust?
spherically symmetric mass
_____
/ - - - \ negative energy
\ + + + / positive energy
--------
• ---->
infinitesimal test mass dm
In the diagram we have an infinitesimal test mass flying fast past a spherically symmetric mass. The test mass probably stretches the radial metric on the far side of the sphere and contracts it on the near side. Pressure does work on the far side but gains energy on the near side. We have marked the areas of energy loss and gain in the diagram.
When test mass moves, it causes energy (or mass) to flow inside the spherical mass. That means that the inertia of the test mass has become bigger.
Let us calculate if the spherical mass has time to adjust to the new metric.
Let us imagine a 1 kg = 10¹⁷ J mass flying past a neutron star whose mass is M ~ 10³⁰ kg and the radius is r = 10 km. The pressure inside the neutron star is p ~ 10³⁴ Pa.
The 1 kg mass stretches the radial metric by a factor
f ~ r_kg / r
= 10⁻³¹,
where r_kg is the Schwarzschild radius 10⁻²⁷ m of a mass of 1 kg.
The pressure drop on the far side is something like
Δp ~ f p
= 1,000 Pa.
The total force pushing the mass M to new positions is
F ~ Δp r²
= 10¹¹ N.
Let the test mass be mildly relativistic and pass the neutron star in a time
t = 10⁻⁴ s.
The mass M of the neutron star can move during that time a distance
d ~ 1/2 F / M * t²
= 5 * 10⁻²⁸ m.
That is about the same as f r = r_kg. We conclude that the neutron star does have time to adjust significantly, even if the test mass is relativistic. If the test mass is slow, then the adjustment is essentially complete.
Let us calculate the figure for an arbitrary star with the same mass M. If we keep M constant
f ~ 1 / r,
p ~ 1 / r⁴,
F ~ 1 / r³,
t ~ r,
d ~ 1 / r.
We see that d is much less than f r = r_kg if r is much larger than 10 km. We conclude that ordinary stars do not have time to adjust to a relativistic test mass. The Sun, or any star, does have time to adjust significantly to a test mass whose velocity is the escape velocity of the star.
Is the metric of general relativity aware of the increase in the inertia?
Is the metric of general relativity aware of this effect? We doubt that.
Suppose that we have a Sun-like star and a relativistic test mass.
When a test mass is flying by, we temporarily, significantly, increase the pressure inside the star. Then the test mass starts to drag with it a larger package of pressure energy inside the star: the inertia of the test mass probably grows because of the extra pressure. The relative change in the inertia of the test mass may be significant, around 10⁻¹⁰ if we double the pressure inside the Sun.
But Birkhoff's theorem says that the metric around a spherically symmetric system cannot change if we change pressure. The inertia cannot change.
Let us analyze the energy flow in the gravity field of the star. The energy of a non-zero gravity field is negative. The test mass makes the field stronger on the far side of the spherical mass. Thus, there is negative energy there.
If we increase the pressure, the gravity field becomes stronger, and there is even more negative energy on the far side. Thus, the energy flow in the gravity field adds more inertia to the test mass. Doubling the pressure of the Sun may add another fraction 10⁻¹⁰ to the inertia of the test mass through the changes in the gravity field.
There may be a general rule that if the interaction between a test charge and a macroscopic system is made stronger, then the test charge acquires more inertia. Increasing the pressure inside the spherical mass makes it more sensitive to the perturbation which the test mass causes. It is no surprise that more pressure means more inertia.
The reaction of the metric to the test mass cannot explain the increased inertia effect
What if we calculate the metric of the spherical mass taking into account its reaction, the pressure changes, caused by the infinitesimal test mass?
Since the equations are nonlinear, we are not able to prove mathematically anything about the reaction.
However, we conjecture that the reaction on the different sides of the spherical mass is infinitesimal. We further conjecture that the metric which we derive from the spherical mass is then only infinitesimally changed. An infinitesimal change in the metric cannot explain a significant relative change in the inertia of the test mass.
What if we replace the spherical mass with a pressurized spherical vessel?
Ehlers et al. (2005) in their paper showed that the calculated static external metric remains the same if we change the pressure in incompressible liquid by tightening a membrane around it. That is, Birkhoff's theorem is satisfied for a static configuration.
Let us have an infinitesimal test mass flying by the vessel.
The pressure inside the liquid does work just like in the case of the spherical mass.
spherical vessel
______
/ - - - \ negative energy
+ | | +
\ + + + / positive energy
----------
• ---->
infinitesimal test mass dm
The negative pressure in the membrane is tangential. Our test mass makes a ring of positive energy to move there as it passes by. In the picture, the + at the equator denotes positive energy in the membrane.
If we tighten the membrane, our test mass drags more pressure energy along.
It probably also drags more negative energy of the gravity field along.
The inertia of the test mass probably increases with pressure.
The variation of a hamiltonian
Suppose that we have an ordinary lagrangian, with no metric. We have a test particle with an infinitesimal charge.
Close to the test charge is a system of macroscopic charges.
Let us assume that we can convert the lagrangian to a hamiltonian.
If we vary the location of the test charge and calculate the difference in the total potential energy, we get the force on the test charge.
The field of the test charge moves along with the test charge. The variation of the position of the test charge by dx involves the variation of its field throughout the space by a small amount.
How do we calculate the inertia of the test charge? We need to find out how much displacement of field energy happens if we move the test charge a distance dx.
This all is complicated.
In the special case of the Coulomb law and electric charges, we can calculate the electric field of the macroscopic charges through a variational principle, and that tells us the force on the test charge. But there is no simple expression for the inertia of the test charge.
For weak electric fields we may assume that the inertia of the test charge is constant, and we may calculate its orbit from the electric field. But if the inertia changes during its orbit, the calculation does not yield the correct orbit.
The self-force of gravity on a test mass
The analysis of the previous sections leaves open the possibility that the metric which the test mass creates around itself could somehow "understand" that the inertia has grown. The question is about the self-force which the gravitational field of the test mass imposes on the test mass itself, and if that force can be expressed with a metric.
The self-force is the obvious mechanism with which the test mass interacts with pressure far away. It is like steel wires attached to the test mass. If the steel wires start to drag along mass-energy from pressure, they slow down the progress of the test mass.
The self-force is private to the test mass. We do not think that one can describe it with a metric which is visible to all observers.
Conclusions
A basic idea of the metric is similar to the electric field: derive a field which we can calculate from the configuration of macroscopic charges, and which determines the orbit of an infinitesimal test charge.
However, taking into account changes in the inertia of a test charge is difficult. An electric field does not claim to know the inertia of a charge at each position of the orbit.
General relativity, on the other hand, claims to know the orbit, which means that it must know the inertia. Can it know it? If it does, that is quite a feat.
We need to check the literature. Has anyone proved that test mass orbits obey the metric?
Few exact solutions of general relativity are known. It is hard to prove mathematically that general relativity does not calculate correct orbits, since we do not know what it calculates, if anything. Demetrios Christodoulou and Sergiu Klainerman (1994) have proved that a small perturbation of the Minkowski metric has a solution in general relativity, but in a more general case we do not know if general relativity has any solutions at all.
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