ivandemartino - Spherically symmetric space-time

Spherically symmetric
space-time

The photon propagation time around a Schwarzschild black hole

The advantage of retaining a fully relativistic approach lies in the ability to describe strong relativistic effects (both on the pulsar orbit and on the photon propagation) in a self-consistent way without the need to introduce the post-Keplerian and post-Newtonian approximations.

This is the approach adopted for the case of a Schwarzschild space-time [1]. Starting from the metric element, the photon travel the time can be written as the integral

depending on the photon path 𝛾 connecting emitter and observer, on the corresponding impact parameter 𝑏, and on the fourth-degree polynomial function R(r). In Schwarzschild space-time, the integral has an exact solution given by

The sign ± depends on whether the integration path is direct or indirect and the functions 𝑇 (𝑟, 𝑏) are, for a specific impact parameter, the results of the integral, given analytically by

where

Here the functions 𝐹, 𝐸 and Π are Jacobian elliptic integrals, the quantities 𝑟1, 𝑟2, 𝑟3, 𝑟4 are the roots of the polynomial 𝑅(𝑟), 𝑥 is an auxiliary variable and 𝑘, 𝑐1 and 𝑐2 are all constant terms built from the roots 𝑟i as shown in Appendix A of [1]. This timing formula has been shown to provide a better description of the photon propagation time with respect to the usual post-Newtonian approximations, being able to capture deviations on the order of seconds when considering a circular orbit with a radius of about a hundred gravitational radii.

📚 We envision, develop, and test a numerical methodology to compute the photon propagation time of the pulses emitted by a pulsar in a generic orbit (and whatever theory of gravity) around a massive compact object, whose gravitational field is described by a spherically symmetric space-time [2].

Numerically solving the emitter-observer problem in a spherically symmetric space-time

Let’s consider a generic asymptotically-flat spherically-symmetric space-time described, in the usual Schwarzschild coordinates (𝑡, 𝑟, 𝜃, 𝜙), by the line element

One can restrict (without loss of generality, due to the spherical symmetry of the problem) to the equatorial plane 𝜃 = 𝜋/2 (which may not coincide with the plane on which the orbit of the emitting object lies), and consider the propagation of light rays on this plane. One can then define the conserved specific (i.e. per unit mass of the test particle) energy, and angular momentum, for a null geodesic. Then, one can define from the conserved quantities the impact parameter which allows rewriting the equations of motion for a null geodesic as

where we have defined

Thus, one can obtain two differential equations directly relating the azimuthal coordinate and the coordinate time to the radial coordinate, respectively:

The latter equation is the one that has to be integrated in order to compute the travel time, and thus the propagation delay, for a photon 𝛾 originating at the emitter position and reaching the observer. In particular, given an emitter located at coordinates 𝐸 (𝑟e, 𝜙e) which emits a photon at coordinate time 𝑡e, and an observer located at 𝑂 (𝑟o, 𝜙o) which receives the photon at coordinate time 𝑡o (see Figure below), we are interested in computing the travel time, Δ𝑡 = 𝑡o − 𝑡e, taken by a photon to cover the distance from 𝐸 to 𝑂. Due to the curved photon path produced by the gravitational field of the central mass, one has to determine the appropriate value for the impact parameter 𝑏 of the photon 𝛾 connecting points 𝐸 and 𝑂 in the curved space-time. This problem is known as the “emitter-observer problem”.

Figure: Illustration of the configuration for the “emitter-observer problem”. The emitter is located at a point 𝐸 that is identified by polar coordinates (𝑟e , 𝜙e), while the observer receiving the photon is located at point 𝑂 with coordinates (𝑟o, 𝜙o). Considering only primary photons received by the observer (i.e. we do not consider photons that graze so close to the unstable photon orbit of the central object that their paths bend so strongly, Δ𝜙 > 2 𝜋, that they reach the observer after one or more complete turns around the central object) only two possible scenarios are possible: (green path) the radial coordinate increases monotonically going from 𝑟e to 𝑟o propagating directly (𝑟e → 𝑟o) from the emitter to the observer; (purple path) the photon leaves the observer with a decreasing radial coordinate (i.e. a negative radial velocity) then reaches a minimum distance 𝑟min from the central object after which it starts increasing again up to the observer position. We call the latter configuration indirect propagation (𝑟e → 𝑟min → 𝑟o).

To solve such a problem, one should obtain the impact parameter by solving the differential equation for the azimuthal coordinate which depends on the photon path 𝛾. Thus one has to take into account whether or not the propagation of 𝛾 from 𝐸 to 𝑂 is direct or indirect (see Figure above for the two different situations). This, in turn, depends on the specific geometrical configuration of the emitter and the observer with respect to the central object (more details in Sect. 3 of [2]). Once the emitter-observer problem has been solved, we have access to the impact parameter 𝑏 of the photon corresponding to the primary image of the source for the observer in 𝑂. We can now approach the problem of determining the travel time of the photon from 𝐸 to 𝑂. Here the assumption of asymptotic flatness of the space-time is crucial for two reasons. First, it allows us to assume that for a sufficiently far observer (i.e. for an Earth-based observer, the distance from SgrA* is ∼ 8 kpc) the curvature of space-time produced by the compact central object can be assumed to be zero at the observer location. While, in the case of the Earth, the presence of the Sun’s gravitational field and the motion of Earth around it can be taken into account using the weak field approximation for the Sun. As a consequence of this, we can assume that the observer actually measures the coordinate time 𝑡, and thus the travel time is simply the integral of the equation of the coordinate time over the photon’s path 𝛾:

Again, as done for the integrals of the angular coordinate, the information on the photon path 𝛾 is encoded in the impact parameter 𝑏 that corresponds to the one resulting from the solution of the emitter-observer problem and in the fact that one either integrates directly from 𝑟e → 𝑟o in the direct propagation case, or passing by 𝑟int (i.e. over the radial path 𝑟e → 𝑟int → 𝑟o) in the indirect one (more details are given in more details in Sect. 3 of [2]).

Time-delay theoretical predictions

Pulsar Timing in the SKA era

Bibliography

Hackmann E., Dhani A., 2019, General Relativity and Gravitation, 51, 37
Della Monica R., De Martino I., De Laurentis, M., "Testing space-time geometries and theories of gravity at the Galactic centre with pulsar's time delay" 2023, Mon. Not. R. Astron. Soc., 524, 3782.
Della Monica R., De Martino I., "Pulsar timing in the Galactic Center", 2025, arXiv:2501.03912

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