Einstein's equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the fo ...
gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
at any point of space at some instant of time t are generated by the mass taken in the preceding (or retarded) instant of time s < t on its world-line at a vertex of the null cone connecting the mass and the field point. The position of the mass that generates the field is called the retarded position and the Liénard–Wiechert potentials are called the retarded potentials. Gravitational waves caused by acceleration of a mass appear to come from the position and direction of the mass at the time it was accelerated (the retarded time and position). The retarded time and the retarded position of the mass are a direct consequence of the finite value of the speed of gravity, the speed with which gravitational waves propagate in space.
As in the case of the
Liénard–Wiechert potential
The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations, these descri ...
s for electromagnetic effects and waves, the static potentials from a moving gravitational mass (i.e., its simple gravitational field, also known as gravitostatic field) are "updated," so that they point to the mass's actual position at constant velocity, with no retardation effects. This happens also for static electric and magnetic effects and is required by Lorentz symmetry, since any mass or charge moving with constant velocity at a great distance, could be replaced by a moving observer at the same distance, with the object now at "rest." In this latter case, the static gravitational field seen by the observer would be required to point to the same position, which is the non-retarded position of the object (mass). Only gravitational waves, caused by acceleration of a mass, and which cannot be removed by a change in a distant observer's
inertial frame
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
, must be subject to aberration, and thus originate from a retarded position and direction, due to their finite velocity of travel from their source. Such waves correspond to electromagnetic waves radiated from an accelerated charge.
Note that for gravitational masses moving past each other in straight lines (or for that matter for electromagnetically charged objects), there is little or no retardation effect on the effect from them, which is mediated by "static" components of the fields. So long as no radiation is emitted, conservation of momentum requires that forces between objects (either electromagnetic or gravitational forces) point at objects' instantaneous and up-to-date positions, and not in the direction of their speed-of-light-delayed (retarded) positions. However, since no information can be transmitted from such an interaction, such influences (which seem to exceed that of the influence of light), cannot be used to violate principles of relativity.
Liénard–Wiechert potential
The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations, these descri ...