**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

(1):

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

(1′):

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

(2):

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:

(2′):

where the straight line itself is

(2.S):

Up to first order in , Synge’s world function is

(2”):

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

(2”’):

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

(2.Synge):

with the straight line already given in eq. (2.S).

**Linearized Einstein’s Equations** If we choose the de Donder gauge

(3):

where we are moving indices with the flat Cartesian metric and

(3′):

Einstein’s field equations linearized about flat spacetime yields

(3”):

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

(3.N):

such that eq. (3”) is now dominated by the Poisson equation

(3”’):

( 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:

(3.NS):

**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:

(4):

(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)

(4′):

Taking the positive square root on both sides, we find that the time elapsed is

(4.Shapiro):

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!

**References**

- I.I. Shapiro, “Fourth Test of General Relativity,” Phys. Rev. Lett.
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Phys. Rev. Lett.**20**, 1265 (1968). doi:10.1103/PhysRevLett.20.1265 - I.I. Shapiro, M.E. Ash, R.P. Ingalls, W.B. Smith, D.B. Campbell, R.B. Dyce, R.F. Jurgens and G.H. Pettengill, “Fourth test of general relativity – new radar result,” Phys. Rev. Lett.
**26**, 1132 (1971). doi:10.1103/PhysRevLett.26.1132 - M.J. Pfenning and E. Poisson, “Scalar, electromagnetic, and gravitational selfforces in weakly curved space-times,” Phys. Rev. D
**65**, 084001 (2002) doi:10.1103/PhysRevD.65.084001 [gr-qc/0012057]. - 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]].