Geodesics Consider all possible spacetime trajectories joining the points and . Suppose you found a trajectory (obeying and ) such that any perturbation away from it yields a slightly longer or shorter path — then such a path is said to extremize the distance between and . In differential geometric/general relativity lingo, such a path is called a geodesic.
In a given spacetime metric and given a geodesic trajectory joining to , half the square of the geodesic distance between and can be written as
where is dubbed “Synge’s World Function” in the literature.
Conversely, if we view eq. (1) as a functional of the trajectory , and we demand it to be extremized, then it would yield the (affinely parametrized) geodesic equation
To sum: Synge’s world function is the action principle for affinely parametrized geodesic motion; and when evaluated on a given geodesic joining to , it hands us half the square of the geodesic distance between this pair of spacetime points.
Perturbation Theory In a weakly curved spacetime of the form
where the components of are assumed to be much less than unity; the Synge’s world function may be used to find an accurate integral solution for geodesic distances merely from the geodesic solutions in flat spacetime — namely, a straight line joining to — precisely because is the geodesic action principle. To see this, we first express the geodesic solution of the geometry in eq. (2) as a perturbation away from a straight line:
where the straight line itself is
Up to first order in , Synge’s world function is
Here, the boundary conditions were used to set to zero the boundary terms; we have used eq. (2.S) to infer ; and, finally, the ‘geodesic operator’ reads
Both and in eq. (2”) must scale as order or higher since they vanish in the limit as . (The arguments for and can be made explicit by direct computation for the former; and, for the latter, by first converting the geodesic equation of eq. (1′) into an integral equation, followed by employing the Born-series-approximation iteration technique.) Therefore the group of terms on the right in eq. (2”) must scale as or higher and may thus be dropped if all we are seeking is a first order accurate expression.
To summarize: at first order in the metric perturbation, half the square of the geodesic distance between the pair of spacetime points and in a weakly curved spacetime (cf. eq. (2)) is given by the integral
with the straight line already given in eq. (2.S).
Linearized Einstein’s Equations If we choose the de Donder gauge
where we are moving indices with the flat Cartesian metric and
Einstein’s field equations linearized about flat spacetime yields
with denoting the flat spacetime wave operator and is the portion of the matter stress-energy tensor that does not contain any metric perturbations. Now, in the non-relativistic limit, stress-energy-momentum is dominated by the energy density; if is some characteristic speed of the internal dynamics of the source (in its rest frame), we usually have and . In such a scenario, we may parametrize the metric perturbation as a unit matrix proportional to the Newtonian potential
such that eq. (3”) is now dominated by the Poisson equation
( is the spacetime dimension.) What’s crucial for the Shapiro delay discussion below is that this Newtonian potential is strictly negative — provided the energy density is positive () — since the Euclidean Green’s function is strictly negative:
Shapiro Delay Consider two observers at spatial locations and sending signals to each other via null rays (e.g., high frequency electromagnetic waves). Suppose the null rays pass through a region of spacetime near an isolated non-relativistic matter source, we may compute the time-of-flight between emission at to reception at using the Synge’s world function. Since null rays are involved, that means the Synge’s world function in eq. (2.Synge) is zero:
(Remember, .) We have used the shorthand for the time elapsed; and for the Euclidean coordinate distance between the observers. We see that, if there were no matter source, so that , the Minkowski light cone condition would be recovered. That in turn means the multiplying the -integral may be replaced with , since the error incurred would be of second order. This leads us to deduce from eq. (4)
Taking the positive square root on both sides, we find that the time elapsed is
We have arrived at the main result: Shapiro time delay. If the matter source were absent, the spacetime would be completely flat and . But now that is non-trivial, we see it increases the time-of-flight because the in eq. (4.Shapiro) is positive; which in turn is due to the positive character of the energy density in eq. (3.NS). In fact, if energy density were strictly negative, , notice this would decrease the time-of-flight and the effective speed of light would be faster than that in flat spacetime!
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- Y. Z. Chu and G.D. Starkman, “Retarded Green’s Functions In Perturbed Spacetimes For Cosmology and Gravitational Physics,” Phys. Rev. D 84, 124020 (2011) doi:10.1103/PhysRevD.84.124020 [arXiv:1108.1825 [astro-ph.CO]].