The Confrontation between General Relativity and Experiment

Alternatives to general relativity - Wikipedia, the free encyclopedia

But what might be a good definition of gravity? Something like this? Some field interaction with these features:

This applies both to standalone theories and to parts of larger theories. Even with these two constraints, physicists have thought up numerous theories that satisfy them.

- Source terms are only functions of energy/momentum densities/fluxes and not other features.
- Acting on matter independent of its composition though ignoring self-gravity (the Weak Equivalence Principle).

The two mainstream ones:

Newtonian gravity.

Galilean-invariant flat space-time with a quasi-static scalar potential related to the mass density.

Field equation:

Equation of motion:

V = potential, ρ = mass density, G = gravitational constant, F = force

It's mathematically equivalent to Newton's formulation, but in the language of present-day field theories.

General relativity.

Locally Lorentz-invariant curved space-time whose curvature is related to the energy-momentum tensor.

Field equation:

Equation of motion:

G = Einstein tensor, obtained from the Riemann curvature tensor, calculated from the space-time metric

T = energy-momentum tensor, which contains the mass/energy and momentum density and flux.

κ = 8πG(Newton)

D = covariant derivative, a partial derivative over space-time with additional terms from the space-time metric

u = 4-velocity:

x = space-time position

τ = object's proper time. A similar parameter can be used for null geodesics.

Newtonian gravity was the first, but it is rather obviously incompatible with special relativity. Albert Einstein recognized that, and his quest for a SR-compatible theory of gravity led to GR. Many alternatives that have been proposed have the behavior of (nongravitational) matter only dependent on the space-time metric, thus yielding the same equation of motion as for GR, and also satisfying criterion (2), the equivalence principle for nongravitational mass.

But restricting oneself to metric theories of gravity still allows numerous possibilities.

One of the more complicated ones is Jacob Bekenstein's TeVeS (tensor-vector-scalar) theory. It is intended to be a relativity-friendly way of implementing MOND (Modified Newtonian Dynamics).

- Constraints on the metric:

- Conformally-flat metric: (some function) * (manifestly flat metric) -- Nordström's and related theories
- Additional metrics -- bimetric theories
- Additional fields:

- Scalar -- Brans-Dicke, etc.
- Vector
- An additional metric may be interpreted as an additional tensor field.

How does one test these varied possibilities? A big problem is the lack of easily accessible very strong gravitational fields, fields with space-time curvature approaching their size scales. So one has to work with weak fields, and expand around the Newtonian limit to get "post-Newtonian" terms. In post-Newtonian order, most alternatives can be collapsed into a "parametrized post-Newtonian" formalism, with ten parameters multiplying various terms in the space-time metric, parameters that one calculates for each theory (Parameterized post-Newtonian formalism - Wikipedia). One then tries to find the PPN parameter values from observations. Here's a super simple version of the PPN metric with most of its terms omitted:

c = 1, V = Newtonian gravitational potential, γ and β are PPN parameters. I've omitted velocity-dependent and pressure-dependent terms and a rather complicated overlap term.

γ = (space distortion) / (time distortion)

β = (nonlinearity in time distortion)

In GR, γ = β = 1 with all other PPN parameters being zero. For theories with a conformally-flat metric, it is easy to show that γ = -1, though β depends on the specific theory.

Observed results?

Deflection of light and delay of radio signals are (1 + γ) * (Newtonian values for travel at c ignoring SR)

Periapsis extra precession is (2 + 2γ - β)/3 * (GR value)

Etc.

For deflection of light, the early solar-eclipse results were not very good. They could rule out the Newtonian value, but were only weakly consistent with the GR value. But recent results on both deflection and delay are much better, finding γ to be 1 to within less than 10^{-4}.

From observing Mercury and Mars, one finds β to be 1 within about 10^{-4}.

Many alternatives make the gravity of an object's self-gravity different from that of its nongravitational mass, while GR makes them equal. If different, it polarizes the orbits in 3-body systems like the Earth and the Moon around the Sun. This "Nordtvedt effect" has been searched for there, and its parameter value has an absolute value less than 10^{-3}.

These results, alongside other experimental limits, rule out all alternatives except for those with adjustable parameters that make them arbitrarily close to GR. Alternatives like scalar-tensor theories, where the scalar field changes the effective value of the gravitational constant. Since one can make that effect arbitrarily small, one can make these theories into GR + extra scalar field.

Gravitational radiation is more complicated to calculate, and in GR, it is post-Newtonian order 2.5, but inspiral of some binary pulsars agrees with GR predictions.

Strong-field tests? Neutron stars have the problem that the equation of state for nuclear matter is not very well-understood above nuclear densities. Black holes have a similar problem, the problem of the physics of accretion disks. These are formed by material orbiting them as it spirals in.

But I think I'll stop here.