GRAVITATION

INTRODUCTION

The light deflection by a massive spherically symmetrical object is a phenomenon predicted by classical mechanics and general relativity.
The two theories, however, give significantly different results.
We develop here, for the photon trajectories in spherical symmetry, a method for analytically calculating the trajectory, parameters and theoretical deflection of the photon within the framework of classical mechanics, and a method for calculating the trajectory parameters and numerically integrating the trajectory and deflection of the photon within the framework of general relativity and Schwarzschild metric.
The differences are highlighted by comparing the two cases, using as numerical examples the sun as a massive object and a hypothetical black hole of the mass of the sun.
Finally, different types of trajectories for a photon arriving from infinity or emitted near a black hole are discussed within the framework of general relativity, leading to some unusual phenomena that depend on the locations of the observer and of the light source.
Note: to avoid any confusion, the writing simplification \(c=G=1\) is not used in the present document and all equations are written explicitly.

CLASSICAL MECHANICS – PHOTON TRAJECTORIES IN SPHERICAL SYMMETRY

Warning: the application of classical mechanics to calculate the photon trajectories deflected by a massive object in spherical symmetry assimilates the photon to a massive particle in order to apply the conservations of angular momentum and mechanical energy. This is done here within a theoretical framework in order to compare these results with those given by general relativity. As the photon has no mass, the results given by classical mechanics are not in line with the observations.

General equation of the trajectory

Considering an isolated system consisting of a massive object of mass \(M\) and a particle \(p\) of mass \(m\), and assuming \(m\ll M\), the center of gravity of the system practically coincides with the center of gravity \(O\) of the massive object.
In a Galilean referential linked to \(O\), the particle \(p\) represented by a point \(P\) subjected to a central force field at the point \(O\) moves in the plane defined by its position vector \(\overrightarrow{OP}\) and its velocity vector \(\vec{v}\), considering the invariance of its angular momentum¹.
By defining an axis \(x\) originating from \(O\), parallel to the initial velocity vector \(\vec{v}\) of \(p\) and of opposite direction, the trajectory of \(p\) can be defined by its polar coordinates (\(r\), \(\varphi\)), where \(r\) is the distance \(OP\) and \(\varphi\) the angle between the axis \(O_x\) and the vector \(\overrightarrow{OP}\) (phase).
Considering the constant \(K\) norm of angular momentum per unit mass, which is \(|r^2\frac{d\varphi}{dt}|\) or \(||\overrightarrow{OP}\land\vec{v}||\), with the assumption \(K\ne 0\) (the opposite case is discussed in paragraph 2.5) and with the change of variable \(u={1\over r}\), the 2\(^{nd}\) order differential equation for calculating the trajectory \(u(\varphi)\) is:
\(\frac{d^2u}{d\varphi^2}+u=\frac{GM}{K^2}\hspace{2cm}\)(2.a),
with \(G\) gravitational constant, \(M\) mass of the massive object and \(K\) defined above.
The general analytical solution of (2.a) is a conic of equation:
\(u(\varphi)=\frac{GM}{K^2}(1+e\cos(\varphi-\varphi_0))\)²\(\hspace{2cm}\)(2.b).
The angle \(\varphi_0\) (conic axis) and eccentricity \(e\) are the two integration constants to be determined from the initial conditions of \(p\): position \(u\) and value of \(K\).
\(p\) is said to be « entering » if \(r\) is decreasing with time (\(\frac{dr}{dt}<0\)) or \(u\) increasing (\(\frac{du}{dt}>0\)), and « exiting » if \(r\) is increasing with time (\(\frac{dr}{dt}>0\)) or \(u\) decreasing (\(\frac{du}{dt}<0\)).

Parameters \(L\) and \(\frac{dL}{d\varphi}\)

The tangent to the trajectory at point \(P\) (\(r\) , \(\varphi\)) is the straight line that goes through \(P\) and which makes an angle \(V\) with the vector \(\overrightarrow{OP}\) such that \(\tan V=r\frac{d\varphi}{dr}\).
The complementary angle \(L\) of the angle \(V\) is defined by \(\tan L=-\cot V={1\over u}\frac{du}{d\varphi}\) and corresponds to the observation elevation with respect to the plane perpendicular to the trajectory plane.
The analytical formulae for \(L\) and \(\frac{dL}{d\varphi}\) are given in classical mechanics appendix A.1.

Particle coming from infinity with non-zero velocity

The invariant \(K\) seen above can be calculated from the initial conditions and its value is then \(xv_y-yv_x=bv_\infty\), with \(b>0\).
Replacing \(K\) by its value in (2.b) gives:
\(u(\varphi)=\frac{GM}{v_\infty^2b^2}(1+e\cos(\varphi-\varphi_0))\hspace{2cm}\)(2.c).
Mechanical energy \(E_{meca}\) = kinetic energy + potential energy is an invariant³ and given that potential energy \(-\frac{GmM}{r}\) is zero for \(r\ \infty\) that is \(u(0)=0\), \(E_{meca}\) is then \({1\over 2}mv_\infty^2\).
Furthermore, \(E_{meca}\) can be written (see classical mechanics appendix A.2):
\(E_{meca}={1\over 2}\frac{(GmM)^2}{mK^2}(e^2-1)\) that is with \(K=v_\infty b\): \(E_{meca}={1\over 2}\frac{(GmM)^2}{mv_\infty^2b^2}(e^2-1)\) which, with the initial value seen above, gives: \({1\over 2}mv_\infty^2={1\over 2}\frac{(GmM)^2}{mv_\infty^2b^2}(e^2-1)\) and after calculation:
\(e=\sqrt{1+\frac{v_\infty^4b^2}{G^2M^2}}\hspace{2cm}\)(2.d).
With \(u(0)=0\), it comes from (2.c): \(\cos \varphi_0=−{1\over e}\) that is:
\(\varphi_0=\arccos\left(-{1\over e}\right)\hspace{2cm}\)(2.e).
If \(v(r)\) is the speed of \(p\) at a distance \(r\), invariance of mechanical energy allows to write: \({1\over 2}mv_\infty^2={1\over 2}mv(r)^2-\frac{GmM}{r}\) that is: \(v(r)=\sqrt{v_\infty^2+\frac{2GM}{r}}\hspace{2cm}\)(2.f).
Considering now that \(p\) is a photon, \(v_\infty=c\) speed of light in vacuum, which gives by application in (2.d), (2.e) and (2.f) and with
\(R_s=\frac{2GM}{c^2}\) (Schwarzschild radius):
\(e=\sqrt{1+\frac{4b^2}{R_s^2}}\hspace{2cm}\)(2.g),
\(\varphi_0=\arccos\left(-\frac{1}{\sqrt{1+\frac{4b^2}{R_s^2}}}\right)\hspace{2cm}\)(2.h),
and \(v(r)=c\sqrt{1+{R_s\over r}}\hspace{2cm}\)(2.i),
which means that in classical mechanics, the speed of the photon is not an invariant.
In this context, the photon trajectories deflected by a massive object in spherical symmetry are then, according to (2.c), defined by:
\(u(\varphi)=\frac{R_s}{2b^2}(1+e\cos(\varphi-\varphi_0))\hspace{2cm}\)(2.j),
with \(e\) and \(\varphi_0\) given by (2.g) and (2.h) respectively,
which shows that the trajectory \(p\) arriving from infinity at speed \(c\) is entirely determined by the values of the impact parameter \(b\) and the mass \(M\) of the massive object (through \(R_s\)).
The derivative \(\frac{du}{d\varphi}\) is \(-\frac{R_s}{2b^2}e\sin(\varphi-\varphi_0)\) which can also be written: \(\frac{du}{d\varphi}=\sqrt{\frac{R_su}{b^2}-u^2+{1\over b^2}}\) for \(\varphi<0\) and \(\frac{du}{d\varphi}=-\sqrt{\frac{R_su}{b^2}-u^2+{1\over b^2}}\) for \(\varphi>0\hspace{2cm}\)(2.k),
and (2.j) gives: \(\frac{d^2u}{d\varphi^2}=-u+\frac{R_s}{2b^2}\hspace{2cm}\)(2.l).
It is easy to check that the trajectories of two photons \(p_1(R_{s_1},b_1)\) and \(p_2(R_{s_2},b_2)\) with \(b_2=b_1\frac{R_{s_2}}{R_{s_1}}\) will be identical to the extent of the homothety factor \(\frac{R_{s_2}}{R_{s_1}}\left(=\frac{M_2}{M_1}\right)\): \(r_2(\varphi)=\frac{R_{s_2}}{R_{s_1}}r_1(\varphi)\left(=\frac{M_2}{M_1}r_1(\varphi)\right)\).
From (2.a), taking \(K=bc\) and \(GM={1\over 2}R_s\ c^2\), this property is transferable, whether the photon comes from infinity or not.
According to (2.g), the eccentricity \(e\) is \(>1\) which indicates that the trajectory of \(p\) is a branch of hyperbola of focus \(O\), symmetrical with respect to the angle \(\varphi_0\).
The mathematical extreme case corresponds to a massive object reduced to a material point of mass \(M\) and an impact parameter \(b=0\) giving an eccentricity \(e=1\) and a parabola reduced to a half straight line with an angle \(\varphi_0=\pi\). \(p\) comes from infinity, reaches the material point \(O\) of mass \(M\) and heads back to infinity in the direction from which it came.
Note that since the photon has no mass, the center of gravity of the isolated system made up of the massive object and the photon is merged with the center of gravity of the massive object. Consequently, in the context of classical mechanics, (2.g) to (2.l) give exact values, while (2.a) to (2.f), which apply to a particle of mass \(m\) give approximate values.

Pericentre and impact parameter \(b\) limit

(2.j) has an absolute maximum \(u\) which is \(u(\varphi_0)=\frac{R_s}{2b^2}(1+e)\).
This value is also found with (2.k) by nullifying \(\frac{du}{d\varphi}\).
The maximum of \(u\) corresponds by definition to the minimum of \(r\) (crossing the pericentre):
\(r_{per}={1\over u(\varphi_0)}=\frac{2b^2}{R_s(1+e)}\hspace{2cm}\)(2.m).
The condition for \(p\) not to meet the massive object is \(r_{per}>R\) radius of the sphere representing the shape of the massive object, that is:
\(\frac{2b^2}{R_s(1+e)}>R\).
Replacing \(e\) by its value given by (2.g), it comes after a simple calculation not detailed here:
\(b>b_{lim}=R\sqrt{1+\frac{R_s}{R}}\) or \(b_{lim}=R\sqrt{1+\frac{2GM}{c^2R}}\hspace{2cm}\)(2.n).
\(p\) comes from infinity and continues towards infinity, having been deflected by the massive object at a minimum distance \(r_{per}\).
Thus, there is always a value \(b=b_{lim}\) that allows the photon to « tangent » the massive object, whatever its mass \(M\) or non-zero radius \(R\).
Furthermore, (2.j) means that the photon trajectories deflected by a massive object in spherical symmetry are symmetrical about the \(\varphi_0\) axis.

Total deflection

Provided that \(p\) does not meet the massive object (\(b>b_{lim}\)), the trajectory is symmetrical with respect to the angle \(\varphi_0\) which allows us with (2.h) to calculate the total deflection of \(p\), from the total angle \(2\varphi_0\) and the angle without deflection \(\pi\):
Total deflection = \(2\arccos\left(-\frac{1}{\sqrt{1+\frac{4b^2}{R_s^2}}}\right)-\pi\hspace{2cm}\)(2.o), with \(R_s=\frac{2GM}{c^2}\)
(2.o) gives the exact total deflection of \(p\) subjected to a gravitational field generated by a massive object of mass \(M\), \(p\) being assumed to come from infinity and continue towards infinity.
Consequently, the deflection of the photon trajectories by a massive object in spherical symmetry according to classical mechanics has a maximum for \(b=b_{lim}\) which corresponds to the minimum value of \(-\frac{1}{\sqrt{1+\frac{4b^2}{R_s^2}}}\) that is \(-\frac{1}{\sqrt{1+\frac{4b_{lim}^2}{R_s^2}}}\) which, replacing \(b_{lim}\) by its value (2.n), leads to:
Maximum total deflection = \(2\arccos\left(-\frac{1}{\sqrt{1+\frac{2R}{R_s}}}\right)-\pi\hspace{2cm}\)(2.p).
In the mathematical extreme case of the material point \(R=0\), the maximum total deflection is \(\pi\).

Approximation of the total deflection

For \(2b\gg R_s\), the total deflection may be approximated.
In this case \(\sqrt{1+\frac{4b^2}{R_s^2}}\simeq\sqrt{\frac{4b^2}{R_s^2}}=\frac{2b}{R_s}\) and (2.o) then gives:
Total deflection \(\simeq 2\arccos\left(-\frac{R_s}{2b}\right)-\pi\hspace{2cm}\)(2.q), with \(\frac{R_s}{2b}\ll1\)
In the vicinity of \(h=0\), the limited expansion of \(\arccos(h)\) to first order gives:
\(\arccos(h)\simeq\frac{\pi}{2}-h\), hence since \(\frac{R_s}{2b}\) is very small:
\(\arccos(-\frac{R_s}{2b})\simeq\frac{\pi}{2}+\frac{R_s}{2b}\)
which finally gives according to (2.q):
Total deflection \(\simeq\frac{R_s}{b}\left(=\frac{2GM}{c^2b}\right)\hspace{2cm}\)(2.r).
The maximum total deviation given by (2.p) may be approximated in the same way for \(\frac{R_s}{R}\ll1\) which finally gives:
Maximum total deflection = \(2\arccos\left(-\frac{1}{\sqrt{1+\frac{2R}{R_s}}}\right)-\pi\simeq\frac{R_s}{R+\frac{R_s}{2}}\left(=\frac{1}{\frac{c^2R}{2GM}+{1\over 2}}\right)\hspace{2cm}\)(2.s),
or with (2.r):
Maximum total deflection \(\simeq\frac{R_s}{b_{lim}}=\frac{R_s}{R\sqrt{1+\frac{R_s}{R}}}\simeq\frac{R_s}{R+\frac{R_s}{2}}\).

Impact

As seen previously, if \(b<b_{lim}\), \(p\) impacts the massive object.
In the case of a compact object of radius \(R_s\), the condition \(b<b_{lim}\) leads according to (2.h):
\(\varphi_0<\arccos\left(-\frac{1}{\sqrt{1+\frac{4b_{lim}^2}{R_s^2}}}\right)\) or by replacing \(b_{lim}\) by its value (2.n):
\(\varphi_0<\arccos\left(-{1\over 3}\right)\simeq 109.5^\circ\)
(2.k) gives for an entering photon and \(\frac{d\varphi}{dt}>0\), \(u={1\over R_s}\) and \(\tan(L)={1\over u}\frac{du}{d\varphi}\):
\(\tan L_{impact}=R_s\sqrt{\frac{2}{b^2}-{1\over R_s^2}}\) or \(L_{impact}=\arctan\sqrt{\frac{2R_s^2}{b^2}-1}\).

Photon emitted by a light source not located at infinity

The trajectory of an entering or exiting photon \(p\) emitted from a light source not located at infinity with phase angle \(\varphi_{em}\) and a distance \(r_{em}>R_s\) is a hyperbolic branch obtained in a similar way to the calculations above, from equation (2.a), the initial conditions at emission, and the invariance of angular momentum and mechanical energy, which allow us to determine its eccentricity and axis, considering that at the initial position (\(r_{em},\varphi_{em}\)) its speed is \(c\) (see classical mechanics appendix A.3 for calculations):
\(u(\varphi)=\frac{R_s}{2b^2\left(1-\frac{R_s}{r_{em}}\right)}(1+e\cos(\varphi-\varphi_0))\)
with \(e=\sqrt{1+\frac{4b^2}{R_s^2}\left(1-\frac{R_s}{r_{em}}\right)^2}\) and
\(\varphi_0=\varphi_{em}+\arccos\left(\frac{\frac{2b^2\left(1-\frac{R_s}{r_{em}}\right)}{r_{em}R_s}-1}{\sqrt{1+\frac{4b^2}{R_s^2}\left(1-\frac{R_s}{r_{em}}\right)^2}}\right)\)
The pericentre \(r_{per}\) is \(\frac{2b^2\left(1-\frac{R_s}{r_{em}}\right)}{R_s(1+e)}\) and for a massive object of radius \(R\), \(b_{lim}\) is calculated by writing \(\frac{2b^2\left(1-\frac{R_s}{r_{em}}\right)}{R_s(1+e)}>R\)
replacing \(e\) by its value and following a simple calculation not detailed here gives:
\(b>b_{lim}=R\sqrt{1+\frac{R_s}{R\left(1-\frac{R_s}{r_{em}}\right)}}\) condition for the photon not to impact the massive object.
Moreover, the invariance of mechanical energy allows us to find the speed of \(p\) function of \(r\):
\({1\over 2}v(r)^2-\frac{GM}{r}={1\over 2}c^2-\frac{GM}{r_{em}}\)
or \(v(r)=c\sqrt{1-\frac{R_s}{r_{em}}\left(1-\frac{r_{em}}{r}\right)}\hspace{2cm}\)(2.t).

Photon emitted from the surface of the massive object

The trajectory of a photon \(p\) exiting the surface of a massive object of mass \(M\) and radius \(R>R_s\) is a branch of a hyperbola, and eccentricity and axis, as well as speed \(v(r)\) of \(p\) are found by replacing \(r_{em}\) par \(R\) in the equations in paragraph 2.4.
At infinity, \(v_\infty=c\sqrt{1-\frac{R_s}{R}}\) (in classical mechanics, the speed of the photon is not an invariant).
If \(R=R_s\), the mechanical energy of \(p\) is zero and \(v_\infty=0\).
If \(R<R_s\), the mechanical energy of \(p\) is < 0, which leads to an eccentricity \(<1\) (see classical mechanics appendix A.2 (2.y)): \(p\) will follow an elliptical orbit around the massive object.

Circular orbit

If the radius \(R\) of the massive object is less than \({R_s\over 2}\) and the photon is emitted at the radial coordinate \(r_{em}={R_s\over 2}\) perpendicularly to the axis joining the emission point and the center \(O\) of the massive object, the eccentricity is zero: \(p\) will follow a stable circular orbit of radius \({R_s\over 2}\) at constant speed \(c\) around the massive object.
In classical mechanics, this is the only case where the photon trajectories are performed with an invariant speed equal to \(c\) (see classical mechanics appendix A.3).

Zero speed at infinity and non-zero angular momentum

Photon emitted with \(r\) emission \(=R_s\)

Zero angular momentum – Radial trajectories

The nullity of \(K=\left|r^2\frac{d\varphi}{dt}\right|\) results in \(r\) and/or \(\frac{d\varphi}{dt}=0\).
The case \(r=0\) is the trivial case where \(p\) is located at the center of gravity \(O\) of the massive object and has no motion.
The case \(\frac{d\varphi}{dt}=0\) means that \(\varphi\) keeps its initial value and the trajectory of \(p\) is on a constant \(\varphi_{initial}\) axis (no deviation) which corresponds to a radial trajectory.
A photon \(p\) entering from infinity (or from a coordinate \(r_{em}>R\) radius of the massive object) on the \(O_x\) axis, with an initial velocity of zero or \(<0\), will remain on this axis by impacting the massive object when \(r\) reaches \(R\) .
A photon \(p\) exiting from a coordinate \(r_{em}>R\) on the \(O_x\) axis, with an initial velocity \(>0\), will remain on this axis as it moves towards infinity.
Invariance of mechanical energy applies and equations (2.i) and (2.t) remain valid, with \(v(r)\) representing the speed of \(p\) along the \(O_x\) axis of the trajectory.

Numerical implementation

Warning: the numerical values calculated below for the photon trajectories deflected by a massive object in spherical symmetry according to classical mechanics are given mathematically with 15 digits, this level of accuracy having no physical meaning. Results with significant digits (linked to the accuracy of the input values \(G\), \(c\), \(M\) et \(R\), and to the calculation formula) are given in brackets.

Photon coming from infinity in the gravitational field of the sun

Assuming that the massive object is the sun, and knowing the values of the constants:
\(G\) gravitational constant \(6.6743\ 10^{-11}m^3.kg^{-1}.s^{-2}\)
\(c\) speed of light in a vacuum \(299\ 792 \ 458\ m.s^{-1}\)
\(M\odot\) mass of the sun \(1.9891\ 10^{30}\ kg\)
\(R\odot\) radius of the sun \(6.96342\ 10^8\ m\)
\(R_s\) calculated Schwarzschild radius \(2\ 953.235415823111\ m\ (2\ 953\ m)\),
the numerical values \(b_{lim}\), \(e\), \(r_{per}\), \(v_{max}\), \(\varphi_0\) and the total deviation are calculated as follows:
(2.n) gives \(b_{lim}=6.963434766161423\ 10^8\ m\ (6.96342\ 10^8\ m)\)
By choosing \(b=b_{lim}\) (case where the photon passes as close as possible to the sun):
(2.g) gives \(e=4.715800663142369\ 10^5\ m\ (4,716\ 10^5\ m)\)
(2.m) gives the pericentre value \(r_{per}=6.96342\ 10^8\ m=R\odot=\) radius of the sun, since \(b=b_{lim}\),
(2.i) can be used to calculate \(v_{max}=v(r_{per})=1.000002120532931\ c\ (1.000002121\ c)\)
(2.h) gives \(\varphi_0=1.5707984473255794\ rad\ (1.57079845\ rad)\),
or an exact total deviation \(=2\varphi_0−\pi=4.241061365650722\ 10^{-6}\ rad\)
\(=0, 8747817008679921\ second\ of\ arc\ (0.875\ second\ of\ arc)\)
\(\frac{R_s}{2b}\) is \(2.12\ 10^{-6}\), and considering this very low value, the use of the approximation (2.r) is justified and gives a total deviation \(\simeq \frac{R_s}{b}=4.2410613655408384\ 10^{-6}\ rad\) which differs from the exact value above only in the \(11^{th}\) digit.
As seen previously (2.p) since \(b=b_{lim}\), the value of the total deviation here is the maximum possible for a photon not absorbed by the sun.

Photon coming from infinity in the gravitational field of a black hole

Assuming that the black hole is equivalent to a sphere of mass \(M\), its physical radius \(R\) is by definition less than or equal to \(R_s=\frac{2GM}{c^2}\).
Taking \(M\) = mass of the sun and \(R=R_s\), the formulae referenced above give:
\(b_{lim}=4\ 176.505577937591\ m\ (4\ 176.5\ m)\).
By choosing \(b=b_{lim}\) (case where the photon passes as close as possible to the black hole):
\(e=3\),
\(r_{per}=2\ 953.235415823111\ m\ (2\ 953\ m)=R_s\) since \(b=b_{lim}\),
\(v_{max}=\sqrt{2}c\),
\(\varphi_0=1.9106332362490186\ rad\ (1.91063324\ rad)\),
that is, an exact total deviation = \(2\varphi_0-\pi=0.6796738189082441\ rad\)
\(=38.9424412689814°\ (38.9^\circ)\), noting that \(\frac{R_s}{2b}\) is \(0.354\) which does not allow approximation (2.r).
Assuming an observer can resist the gravity generated by the black hole and is looking in the opposite direction to it, the following phenomena will appear depending on the position of the observer:

Observer located at a distance \(r\)

At a suitable distance from the black hole (about \(r>5\ 900\ m\)), the observer would see each of the stars located very precisely at the elevation \(−90^\circ\) (that is, on the axis joining the observer and the center of the black hole), in the shape of a very thin luminous circle centered on this axis that could be named « Newton circle© » (equivalent to the « Einstein ring » mentioned later in general relativity).

Observer located at a distance \(r=R_s\)

The analytical calculation shows that the observer would see visible stars whose true elevation lies between \(90^\circ\) and \(−19.5^\circ\) approximately (\(90^\circ-\arccos\left(-{1\over 3}\right)\)), see paragraph 2.3.4. above), the « contraction » factor being slight for elevations close to \(90^\circ\) (\(1.17\) approximately), and slightly more accentuated for elevations close to \(0^\circ\) (\(1.22\) approximately).
This phenomenon also applies to the apparent diameters of the stars, which are smaller than the actual diameters with a maximum contraction factor of \(4\over 3\) for the elevation \(0^\circ\).
See classic mechanical appendix A.5 for more details.

Photon emitted with \(r\) emission \(=R_s\)

(2.t) shows that \(v_\infty=0\) (the speed of the photon cancels out at infinity).
When the photon is emitted tangentially (\(L_{em}=0\)), the parabolic trajectory is also that of a photon entering at zero speed at infinity with \(K=cR_s=\frac{2GM}{c}\) (see classical mechanics appendix A.4).
The total deflection is \(\frac{\pi}{2}\) for the photon exiting from \(r_{em}=R_s\) to infinity and \(\pi\) for the photon entering from infinity and exiting to infinity.
This deflection is greater than that seen in paragraph 2.3.2 above, which assumes that the photon has a speed \(c\) at infinity.

GENERAL RELATIVITY – PHOTON TRAJECTORIES IN SPHERICAL SYMMETRY

General equation of the trajectory

The photon is subjected to a gravitational field generated by a massive object of mass \(M\).
Assuming that the gravitational field is spherically symmetrical, and using the coordinate system (\(ct,\ r,\ \varphi,\ \theta\)), the scalar product \(\overrightarrow{dP}.\overrightarrow{dP}\) of an elementary move \(\overrightarrow{dP}\) of the photon is defined by the Schwarzschild metric tensor (see its limits in the conclusion) and is written:
\(g(dP,dP)=g_{\alpha\beta}dx^{\alpha}dx^{\beta}\)
\(=-\left(1-\frac{R_s}{r}\right)c^2dt^2+\left(\frac{1}{1-\frac{R_s}{r}}\right)\ dr^2+r^2d\theta^2+r^2\sin^2\theta\ d\varphi^2\)⁴\(\hspace{2cm}\)(3.a).
with \(g_{\alpha\beta}\) coefficients of the tensor matrix, \(dx^{\alpha}\) and \(dx^{\beta}\) elementary move coordinates (\(cdt\), \(dr\), \(d\varphi\) or \(d\theta\)), \(c\) speed of light in vacuum (invariant) and \(R_s=\frac{2GM}{c^2}\) (Schwarzschild radius).
In the asymptotic region \(r\gg R_s\), the coordinate \(r\) is interpreted as the physical distance between the photon and the center \(O\) of the massive object.
The symmetries of the Schwarzschild metric imply that the photon trajectories deflected by a massive object in spherical symmetry remain in a plane, chosen to be \(\theta=\frac{\pi}{2}\), which gives:
\(g(dP,dP)=-\left(1-\frac{R_s}{r}\right)\ c^2dt^2+\left(\frac{1}{1-\frac{R_s}{r}}\right)\ dr^2+r^2d\varphi^2\hspace{2cm}\)(3.b).
\(\varphi\) is then the phase of the photon moving in the plane \(\theta=\frac{\pi}{2}\) with its Cartesian reference frame \(O_{xy}\).
The case \(d\varphi=0\) corresponds to a radial trajectory (\(\varphi=const\)) and is discussed in paragraph 3.6.
The four-momentum of the photon can be defined by a vector \(\overrightarrow{p}\) with four coordinates (\(p^0,\ p^r,\ p^\varphi,\ p^\theta\)) in the Schwarzschild coordinate system, with \(p^\theta=0\) since the motion of the photon takes place in the plane \(\theta=\frac{\pi}{2}\).
The quantities \(\varepsilon\) and \(l\) defined as follows:
\(\varepsilon=c\left(1-\frac{R_s}{r}\right)\ p^0\hspace{2cm}\)(3.c),
\(l=r^2p^\varphi\hspace{2cm}\)(3.d),
which are for \(r\gg R_s\) respectively the energy and angular momentum of the photon measured by a static observer (at fixed \(r,\ \varphi\) and \(\theta\)) are invariant along the geodesic of the photon⁵.
For \(r\gg R_s\), the angular momentum \(l\) is also written \(r^2\frac{d\varphi}{dt}\): the case \(l=0\) which corresponds to \(d\varphi=0\) is discussed in paragraph 3.6.
Moreover, the four-momentum \(\overrightarrow{p}\) of the photon is a light-like vector, so its scalar product \(\overrightarrow{p}.\overrightarrow{p}\) is zero⁶. It can be written as:
\(g(p,p)=g_{00}(p^0)^2+g_{rr}(p^r)^2+g_{\varphi\varphi}(p^\varphi)^2=0\hspace{2cm}\)(3.e),
with the coefficients \(g_{00}\), \(g_{rr}\) and \(g_{\varphi\varphi}\) given by (3.b).
(3.c) and (3.d) give \((p^0)^2=\left(\frac{1}{1-\frac{R_s}{r}}\right)^2\frac{\varepsilon^2}{c^2}\) and \((p^\varphi)^2=\frac{l^2}{r^4}\), and it comes with (3.e):
\((p^r)^2=\frac{\varepsilon^2}{c^2}-\frac{l^2}{r^2}\left(1-\frac{R_s}{r}\right)\),
or by setting \(b^2=\frac{c^2l^2}{\varepsilon^2}\), with \(b\neq 0\) (see paragraph 3.6 for the case \(b=0\)):
\((p^r)^2=l^2\left({1\over b^2}-{1\over r^2}\left({1-\frac{R_s}{r}}\right)\right)\hspace{2cm}\)(3.f).
By introducing an affine parameter \(\lambda\) along the light geodesic such that:
\(\overrightarrow{p}=l\frac{\overrightarrow{dP}}{d\lambda}\)⁷\(\hspace{2cm}\)(3.g),
and with \(\overrightarrow{dP}=(cdt,\ dr,\ d\varphi,\ 0)\), the three non-zero coordinates of the four-momentum are: \(p^0=lc\frac{dt}{d\lambda}\), \(p^r=l\frac{dr}{d\lambda}\) and \(p^\varphi=l\frac{d\varphi}{d\lambda}\).
With (3.f) comes: \(\left(\frac{dr}{d\lambda}\right)^2={1\over b^2}-{1\over r^2}\left(1-\frac{R_s}{r}\right)\) and with (3.d): \(\left(\frac{d\varphi}{d\lambda}\right)^2={1\over r^4}\), that is:
\(\left(\frac{d\varphi}{dr}\right)^2={1\over r^4}\frac{1}{{1\over b^2}-{1\over r^2}\left(1-\frac{R_s}{r}\right)}\), which gives:
\(\frac{d\varphi}{dr}=\pm\frac{1}{r^2\sqrt{{1\over b^2}-{1\over r^2}\left(1-\frac{R_s}{r}\right)}}\hspace{2cm}\)(3.h),
which will enable us to calculate the photon trajectories deflected by a massive object in spherical symmetry according to the Schwarzschild metric.

Meaning of \(b\)

With, as seen above \(p^0=lc\frac{dt}{d\lambda}\), \(\varepsilon\) is written according to (3.c):
\(\varepsilon=c^2l\left(1-\frac{R_s}{r}\right)\frac{dt}{d\lambda}\)
hence \(b=\frac{d\lambda}{dt}{1\over c}\left(\frac{1}{1-\frac{R_s}{r}}\right)\) by replacing \(\varepsilon\) by its value in \(b^2=\frac{c^2l^2}{\varepsilon^2}\).
By writing \(\frac{d\lambda}{dt}=\frac{\frac{d\varphi}{dt}}{\frac{d\varphi}{d\lambda}}\), then we have \(b=\frac{r^2}{c}\frac{d\varphi}{dt}\left(\frac{1}{1-\frac{R_s}{r}}\right)\simeq\frac{r^2}{c}\frac{d\varphi}{dt}\) for \(r\gg R_s\hspace{2cm}\)(3.i).
On the other hand, if the photon is emitted in the direction \(x\) with \(r\gg R_s\), the trajectory of the photon is initially at constant \(y\) = impact parameter of the photon with respect to the massive object, which is written \(y_{em}=r\sin\varphi\simeq r\varphi\) for \(r\gg R_s\), which gives \(\varphi\simeq\frac{y_{em}}{r}\) or \(\frac{d\varphi}{dt}\simeq-{1\over r^2}\frac{dr}{dt}y_{em}\).
Taking \(\frac{dr}{dt}\simeq -c\), we get \(\frac{d\varphi}{dt}\simeq\frac{c}{r^2}y_{em}\) and (3.i) then gives \(b=y_{em}\).
\(b\) is therefore the impact parameter for a photon coming from infinity.

In the following, \(p\) refers to the photon.
With the change of variable \(u={1\over r}\), \(\frac{du}{dr}=-u^2\) and (3.h) becomes:
\(\frac{\left(\frac{d\varphi}{dr}\right)^2}{\left(\frac{du}{dr}\right)^2}=\frac{u^4}{\left({1\over b^2}-u^2(1-R_su)\right)u^4}\) or: \(\left(\frac{d\varphi}{du}\right)^2=\frac{1}{R_su^3-u^2+{1\over b^2}}\) that is:
\(\frac{du}{d\varphi}=\pm\sqrt{R_su^3-u^2+{1\over b^2}}\hspace{2cm}\)(3.j),
equation that enables to calculate the photon trajectories deflected by a massive object in spherical symmetry and according to the Schwarzschild metric.
Equation (3.j) shows, on the one hand, that the trajectory of \(p\) coming from infinity (\(u=0\)) is entirely determined by the values of the parameter \(b\) and the mass \(M\) of the massive object (through \(R_s\)) and, on the other hand, that for a given emission point there are two possible geodesics for the photon, corresponding to \(\frac{du}{d\varphi}_{initial}>0\) or \(\frac{du}{d\varphi}_{initial}<0\).
Note: in the case where \(p\) is emitted by a light source not located at infinity, (3.j) implies that the impact parameter \(b\) cannot exceed a specific value (see paragraph 3.4).
\(p\) is said to be « entering » if \(r\) is decreasing with time (\(\frac{dr}{dt}<0\)) or \(u\) increasing (\(\frac{du}{dt}>0\)), and « exiting » if \(r\) is increasing (\(\frac{dr}{dt}>0\)) or \(u\) decreasing (\(\frac{du}{dt}<0\)).
The numerical integrations used to calculate \(u(\varphi)\) are discussed in paragraph 3.7.
If it exists, the extremum of \(r\) is given by writing \(\frac{dr}{d\varphi}=0\) that is, according to (3.h):
\(\frac{r^3}{b^2}-r+R_s=0\hspace{2cm}\)(3.k).
The solving of this equation in the context of (3.h) and the discussion of the photon trajectories deflected by a massive object in spherical symmetry and according to the Schwarzschild metric are detailed in general relativity appendix B.2. The following paragraphs summarize the results, which depend in particular on the value of \(b\) with respect to a critical value:
\(b_{crit}=\frac{3\sqrt{3}}{2}R_s\left(=\frac{3\sqrt{3}\ GM}{c^2}\right)\hspace{2cm}\)(3.l).

Parameters \(L\) and \(\frac{dL}{d\varphi}\)

The geometric definitions of the angle \(V\) of the tangent to the trajectory and its complement \(L\) (see paragraph 2.2 above) apply:
\(\tan V=r\frac{d\varphi}{dr}\) and \(\tan L=-\cot V={1\over u}\frac{du}{d\varphi}\).
The analytical expressions for \(L\) and \(\frac{dL}{d\varphi}\) are given in general relativity appendix B.1.

Photon coming from infinity

Pericentre and impact parameter \(b\) limit

Total deflection

Provided that \(p\) does not meet the massive object (\(b>b_{lim}\)), the total angle is:
\(2\varphi_0=2\int_{r_{per}}^{\infty}\frac{1}{r^2\sqrt{\frac{1}{b^2}-\frac{1}{r^2}\left(1-\frac{R_s}{R}\right)}}dr\)⁸\(\hspace{2cm}\)(3.o),
which gives with the angle without deviation \(\pi\):
Total deflection = \(2\int_{r_{per}}^{\infty}\frac{1}{r^2\sqrt{\frac{1}{b^2}-\frac{1}{r^2}\left(1-\frac{R_s}{R}\right)}}dr-\pi\hspace{2cm}\)(3.p),
for the photon trajectories deflected by a massive object in spherical symmetry and according to the Schwarzschild metric.

Approximation of the total deflection

In the case where the radius \(R\) is \(\gg R_s\), since according to (3.n) \(b>R\), \(b\) is \(\gg R_s\) and a limited expansion in \(\frac{R_s}{b}\) of the integrator of (3.p) gives:
\(2\varphi_0\approx\pi+\frac{2R_s}{b}\)⁹\(\hspace{2cm}\)(3.q),
giving with the angle without deflection \(\pi\):
Total deflection \(\simeq\frac{2R_s}{b}\left(=\frac{4GM}{c^2b}\right)\hspace{2cm}\)(3.r).

Apparent radius of a massive object – Shadow

The apparent radius of a massive object with radius \(R\ge {3\over 2}R_s\) is \(b_{lim}\) since no photon with impact parameter \(b<b_{lim}\) can reach the observer.
If the massive object has a radius \(R\le {3\over 2}R_s\), its apparent radius is \(b_{crit}\) (or \(\frac{3\sqrt{3}}{2}R_s\)) since no photon of impact parameter \(b<b_{crit}\) can reach the observer. It does not depend on \(R\).
No star can be seen through the disk corresponding to the apparent radius (greater than \(R\)) which is then named « shadow » of the massive object.

Circular orbit

Impact

Photon emitted from a light source not located at infinity

Photon emitted from a light source with radial coordinate \(\in [R,\ {3\over 2}R_s]\)

The compact object has a mass \(M\) and its radius \(R\) is assumed to be between \(R_s\) and \({3\over 2}R_s\).

Apocentre and impact

If \(b_{crit}<b\le b_{max}\) the maximum of \(r\) is:
\(r_{apo}=\frac{2b}{\sqrt{3}}\cos\left({1\over 3}\arccos\left(-\frac{b_{crit}}{b}\right)+\frac{4\pi}{3}\right)\hspace{2cm}\)(3.t).
The radial coordinate of \(p\) increases up to the value \(r_{apo}\) then decreases and \(p\) impacts the compact object.
As stated previously in paragraph 3.3.1, (3.j) means that the photon trajectories deflected by a massive object in spherical symmetry are symmetrical with respect to the value of the angle \(\varphi\) at the apocentre (which cancels out \(\frac{du}{d\varphi}\)).
In the case of a black hole, (3.j) gives for \(p\) entering the event horizon and \(\frac{d\varphi}{dt}>0\): \(\frac{du}{d\varphi}={1\over b}\) that is \(\tan L_{impact}=\frac{R_s}{b}\).
The condition \(b>b_{crit}\) then leads to:
\(\tan L_{impact}<\frac{R_s}{b_{crit}}\) or by replacing \(b_{crit}\) by its value (3.l):
\(\tan L_{impact}<\frac{2}{3\sqrt{3}}\) or else \(L_{impact}<L_{crit}=\arctan\left(\frac{2}{3\sqrt{3}}\right)\simeq 21.1^\circ\).

Circular orbit

Release

If \(b<b_{crit}\), (3.k) has no positive root, which means that \(r\) has no maximum and that \(p\) moves away from the compact object towards infinity, with no condition on the value of \(R\).
A photon emitted from the event horizon \(r=R_s\) can therefore escape under the condition \(b<b_{crit}\).

Zero angular momentum at infinity – Radial trajectories

\(d\varphi=0\) (zero angular momentum at infinity) means that the trajectories are not deflected (radial trajectories) and (3.b) becomes for a light geodesic:
\(-\left(1-\frac{R_s}{r}\right)\ c^2dt^2+\left(\frac{1}{1-\frac{R_s}{r}}\right)\ dr^2=0\hspace{2cm}\)(3.u).
(3.u) gives \(c^2dt^2=\frac{dr^2}{\left(1-\frac{R_s}{r}\right)^2}\) or
\(cdt=\pm\frac{dr}{1-\frac{R_s}{r}}=\pm\frac{rdr}{r-R_s}=\pm\frac{(r-R_s+R_s)\ dr}{r-R_s}=\pm\left(1+\frac{R_s}{r-R_s}\right)dr\)
or after integration for \(r>R_s\):
1) if \(\frac{dr}{dt}<0\) (entering photon): \(ct=-\ r-R_s\ln(r-R_s)+const\),
2) if \(\frac{dr}{dt}>0\) (exiting photon): \(ct=r+R_s\ln(r-R_s)+const\),
with \(const\) values to be determined according to the initial conditions.
A photon \(p\) entering on the \(O_x\) axis from infinity (or from a radial coordinate \(r_{em}>R_s\) will have an initial speed \(<0\) and will remain on the axis, impacting the compact object when \(r\) reaches \(R\) radius of the object .
A photon \(p\) exiting on the \(O_x\) from a radial coordinate \(r_{em}>R\ge R_s\) will have an initial speed \(>0\) and will remain on the axis on its way to infinity.

Numerical integrations

(3.j) allows us to express \(\frac{d\varphi}{du}\), and \(\varphi(u)\) can then be calculated analytically by an elliptic integral of \(\frac{d\varphi}{du}\) in the event that there is a pericentre, which is not always the case depending on the value of \(b\) as seen previously. Moreover, the case of a photon emitted at a radial coordinate very close to \(R_s\) cannot be handled by this method.
The numerical integration of \(\frac{du}{d\varphi}\) given by (3.j) is nevertheless possible with, for \(b\ge b_{crit}\) a tricky handling of the change in sign of \(\frac{du}{d\varphi}\) when \(r\) reaches \(r_{per}\) or \(r_{apo}\).
However, by writing:
\(\left(\frac{du}{d\varphi}\right)^2=R_su^3-u^2+{1\over b^2}\hspace{2cm}\)(3.v),
the 3rd-order differential equation (3.v) which gives the photon trajectories deflected by a massive object in spherical symmetry and according to the Schwarzschild metric, can be processed by preventing the singularity due to the change in sign of \(\frac{du}{d\varphi}\).
The derivation of (3.v) then gives: \(2\frac{du}{d\varphi}\frac{d^2u}{d\varphi^2}=3R_su^2\frac{du}{d\varphi}-2u\frac{du}{d\varphi}\) or:
\(\frac{d^2u}{d\varphi^2}={3\over 2}R_su^2-u\hspace{2cm}\)(3.w).
The differential equation (3.w) can then be solved numerically with the initial conditions and overcome the limits of the elliptic integral of \(\frac{d\varphi}{du}\) seen previously.
Moreover, it is easy to check that according to Schwarzschild metric, the photon trajectories deflected by a massive object in spherical symmetry \(p_1(R_{s_1},b_1)\) and \(p_2(R_{s_2},b_2)\) with \(b_2=b_1\frac{R_{s_2}}{R_{s_1}}\) are kept by applying the scaling factor \(\frac{R_{s_2}}{R_{s_1}}\left(=\frac{M_2}{M_1}\right)\): \(r_2(\varphi)=\frac{R_{s_2}}{R_{s_1}}r_1(\varphi)\left(=\frac{M_2}{M_1}r_1(\varphi)\right)\).

Double integration – 4th-order Runge-Kutta

(3.w) is a second-derivative equation that can be numerically integrated as follows by a 4th-order Runge-Kutta method¹⁰:
1) with \(k_1=\frac{d^2u}{d\varphi^2}(u(\varphi))\), \(k_2=\frac{d^2u}{d\varphi^2}\left(u(\varphi)+\frac{\delta \varphi}{2}\frac{du}{d\varphi}\right)\), \(k_3=\frac{d^2u}{d\varphi^2}\left(u(\varphi)+\frac{\delta \varphi}{2}\frac{du}{d\varphi}+\left(\frac{\delta \varphi}{2}\right)^2\ k_1\right)\) and \(k_4=\frac{d^2u}{d\varphi^2}\left(u(\varphi)+\delta \varphi\ \frac{du}{d\varphi}+\frac{\left(\delta \varphi\right)^2}{2}\ k_2\right)\),
the value of \(\frac{du}{d\varphi}\) at the next step is:
\(\frac{du}{d\varphi}\left(\varphi+\delta \varphi\right)=\frac{du}{d\varphi}(\varphi)+\frac{\delta \varphi}{6}(k_1+2k_2+2k_3+k_4)\), and

2) with \(l_1=\frac{du}{d\varphi}(u(\varphi))\), \(l_2=\frac{du}{d\varphi}\left(u(\varphi)+\frac{\delta \varphi}{2}l_1\right)\), \(l_3=\frac{du}{d\varphi}\left(u(\varphi)+\frac{\delta \varphi}{2}l_2\right)\) and \(l_4=\frac{du}{d\varphi}(u(\varphi)+\delta \varphi\ l_3)\),
the value of \(u\) at the next step is:
\(u(\varphi+\delta \varphi)=u(\varphi)+\frac{\delta \varphi}{6}(l_1+2l_2+2l_3+l_4)\)
which is written with \(l_2=\frac{du}{d\varphi}(\varphi)+\frac{\delta \varphi}{2}k_1\), \(l_3=\frac{du}{d\varphi}(\varphi)+\frac{\delta \varphi}{2}k_2\), \(l_4=\frac{du}{d\varphi}(\varphi)+\delta \varphi\ k_3\) and after expansion:
\(u(\varphi+\delta \varphi)=u(\varphi)+\delta \varphi\frac{du}{d\varphi}(\varphi)+\frac{(\delta \varphi)^2}{6}(k_1+k_2+k_3)\)
The calculation step \(\delta \varphi\) can be chosen to be either constant or adaptive, depending on the required accuracy for the radial coordinate \(r\) \(\left(={1\over u}\right)\) of the photon trajectories deflected by a massive object in spherical symmetry.

Initial conditions

The initial conditions for \(\frac{du}{d\varphi}\) and \(u\) in a stationary reference frame \(O_{xy}\) independent of the initial direction of the photon are determined as follows, from the emission coordinates (\(r_{em}\) , \(\varphi_{em}\)) of the photon (\(u_{em}={1\over r_{em}}\)), and two possible cases for \(\frac{du}{d\varphi}\):
1) entering photon \(\left(\frac{du}{dt}>0\right)\) and \(\frac{d\varphi}{dt}>0\) or exiting photon \(\left(\frac{du}{dt}<0\right)\) and \(\frac{d\varphi}{dt}<0\): \(\frac{du}{d\varphi}=+\sqrt{R_su_{em}^3-u_{em}^2+\frac{1}{b^2}}\)
2) exiting photon \(\left(\frac{du}{dt}<0\right)\) and \(\frac{d\varphi}{dt}>0\) or entering photon \(\left(\frac{du}{dt}>0\right)\) and \(\frac{d\varphi}{dt}<0\): \(\frac{du}{d\varphi}=-\sqrt{R_su_{em}^3-u_{em}^2+\frac{1}{b^2}}\)
If the photon comes from infinity, \(u_{em}=0\), it is entering \(\left(\frac{du}{dt}\right)>0\) and the two possible cases become:
1) \(\frac{du}{d\varphi}={1\over b}\)
2) \(\frac{du}{d\varphi}=-{1\over b}\)
Note: according to the equality of (3.i), if in the stationary reference frame \(O_{xy}\) the angle \(\varphi\) increases with the time \(t\) of the static observer then \(\frac{d\varphi}{dt}=\frac{cb}{r^2}\left(1-\frac{R_s}{r}\right)\) or decreases then \(\frac{d\varphi}{dt}=-\frac{cb}{r^2}\left(1-\frac{R_s}{r}\right)\), which means that all along the trajectory of the photon, the angle \(\varphi\) is constantly increasing or constantly decreasing.

Numerical implementation

Warning: the numerical values calculated below for the photon trajectories deflected by a massive object in spherical symmetry and according to the Schwarzschild metric, are given mathematically with 15 digits, this level of accuracy having no physical meaning. Results with relevant numbers (linked to the accuracy of the input values \(G\), \(c\), \(M\) and \(R\) and to the calculation formula) are given in brackets.
\(G\) gravitational constant \(6.6743\ 10^{-11}\ m^3.kg^{-1}.s^{-2}\)
\(c\) speed of light in a vacuum \(299\ 792\ 458\ m.s^{-1}\)
\(M\odot\) masse of the sun \(1.9884\ 10^{30}\ kg\)
\(R_s\) calculated Schwarzschild radius \(=2\ 953.235415823111\ m\) (\(2\ 953\ m)\)
\(b_{crit}\) calculated \(=\frac {3\sqrt {3}}{2}=7\ 672.730680376143\ m\) (\(7\ 673\ m)\)
\(r_{crit}\) calculated \(={3\over 2}R_s=4\ 429.853123734667\ m\) (\(4\ 430\ m)\)

Photon coming from infinity in the gravitational field of the sun

With \(R\odot\) radius of the sun \(6.96342\ 10^8\ m\),
(3.n) gives \(b_{lim}=6.963434766224048\ 10^8\ m\) (\(6.96343\ 10^8\ m)\).
By choosing \(b=b_{lim}\) (when the photon crosses as close to the sun as possible),
(3.m) gives the pericentre value \(r_{per}=6.96342\ 10^8\ m\) which is \(R\odot\) the radius of the sun.
\(\frac{R_s}{b}\) is \(4.24\ 10^{-6}\), and considering this very low value, the use of the approximation (3.q) is justified and gives:
\(\varphi_0\simeq 1.570800567856262\ rad\) (\(1.570800568\ rad\)).
(3.r) gives total deviation \(\simeq 8.482122731005393\ 10^{-6}\ rad=1.7495634016749193\ second\ of\ arc\) (\(1.750\ second\ of\ arc\)), which is twice the value calculated using classical mechanics (see paragraph 2.6.1 above).
Note: to the accuracy of measurement, the photographs of the sun disk vicinity taken by Arthur Eddington and his team during the total eclipse on Principe Island on May 29 1919 confirmed this value (which is twice the value of the classical mechanics theory calculated by (2p)).

Photon coming from infinity in the gravitational field of a compact object

Note: this paragraph considers a compact object with the mass of the sun and a physical radius \(R<r_{crit}\).

Impact parameter \(b>b\) critical

The photon \(p\) comes from infinity and continues to infinity after being deflected by the compact object. The value of the deflection is determined by the value of \(b\).
By way of example, with the integration accuracy of (3.w):
a total deflection of \(\frac{\pi}{2}\) is given with \(b=9\ 107\ m\) and (3.m) gives the value to the pericentre \(r_{per}=6\ 880\ m\).
Note: \(\frac{R_s}{b}\) is \(0.325\), which does not justify the approximation (3.r).
A total deflection of \(\pi\) (or ½ revolution) is given with \(b=7\ 910\ m\), \(r_{per}=5\ 199\ m\).
An observer located on a star with this value of impact parameter would see its own star in the direction of the compact object.
A total deflection of \(\frac{3\pi}{2}\) is given with \(b=7\ 720\ m\), \(r_{per}=4\ 738\ m\).
A total deflection of \(2\pi\) (or 1 revolution around the compact object) is given with \(b=7\ 682.4\ m\), \(r_{per}=4\ 563\ m\).
See figure 3 above for the plot of these 4 trajectories.
A total deflection of \(4\pi\) (or 2 revolutions) is given with \(b=7\ 672.75\ m\) (or about \(b_{crit}+2\ cm\)), \(r_{per}=4\ 435.4\ m\).
See figure 4 above for plot of the trajectory.
At last, a total deviation of \(8\pi\) (or 4 revolutions) is given with \(b\simeq b_{crit}+7\ 10^{-8}\ m\) and \(r_{per}=4\ 429.864\ m\) (or about \(r_{crit}+1\ cm\)).
Note: eliminating the modulo \(2\pi\), the apparent maximum value of the deflection of the photon is \(\pi\).

Impact parameter \(b=b\) critical

\(p\) follows an unstable circular orbit with radial coordinate \(r_{crit}\simeq 4\ 429.853\ m\).
This is the minimum value below which the photon is absorbed by the compact object (see paragraph below).
See figure 5 above for the plot of this trajectory.

Impact parameter \(b<b\) critical

\(p\) impacts the compact object.
A \(\varphi\) variation of \(\pi\) at impact is given for \(b=6\ 582\ m\).
See figure 6 above for the plot of this trajectory.

Interpretation of results and phenomena observable near a black hole (physical \(R\) \(< R_s\))

Applying the Schwarzschild metric to the photon trajectories deflected by a black hole in spherical symmetry, and excluding geodesics involving one or more revolutions of the photons around the black hole, the deflection by the black hole of the light emitted by a star will give two images of the star at a given observation point, corresponding to the two possible photon geodesics, see paragraph 3.1 (3.j) above. If the observer is not on the axis connecting the star to the center of the black hole, the geodesics will have a different impact parameter \(b\).
See figure 7 above for the plots of the two trajectories (with \(b\) calculated for deflections \(\frac{\pi}{2}\) and \(\pi\), or ¼ and ½ revolution), connecting two given points A and B.
Generally speaking, the deflection by a black hole of the light emitted by each visible star will produce two distinct images at opposite azimuths at a given observation point, one with positive elevation and the other with negative elevation. The image with the highest apparent angle (the greatest deflection) will be the one located at the azimuth opposite that of the actual position of the star with respect to the observer. Each revolution of the photons around the black hole will duplicate the two images described above, with the absolute value of the elevation decreasing with the number of revolutions.

Assuming that an observer can approach a black hole and resist its gravity, and look in the opposite direction to it, the following phenomena will depend on the location of the observer:

Observer located at the radial coordinate \(r\) critical \(={3\over 2}R_s\)

The observer will see all the visible stars in the universe gathered in the hemisphere above him.
Furthermore, since the stars each have different directions of observation, the circular orbits around the black hole of the photons emitted by these stars create a spherical surface of radius \(r_{crit}\), hereafter called the « sphere of photons ».
The whole sphere of photons is not observable by definition, but its effects are: an observer located at the radial coordinate \(r_{crit}\) will see all around him at the elevation \(0^\circ\) a thin, very luminous line corresponding to the photons described above and incoming to him.
Since the orbit of the photons on the sphere is unstable, as seen in paragraphs 3.3.5 and 3.5.2 above, photons may escape (without or after one or more revolutions around the black hole) and reach the observer. In this case, this indirect image of the sphere of photons consists of several circles with radii whose minimum limiting value is \(b_{crit}\).
Observer aboard a spaceship
If the spaceship of the observer had its longitudinal axis tangent to the sphere of photons, with its headlights beaming forward, the observer looking forward would see a highly distorted image of the rear of the spaceship (width highly compressed and height highly stretched, with an up/down inversion), the angular height of the image being \(\arctan\left(\frac {spaceship\ height}{spaceship\ length}\right)\) (the shorter the spaceship, the greater the height of its image).
The \(n^{th}\)-order images corresponding to \(n\) revolution(s) of the photons around the black hole will be very thin vertical segments of the same angular height, superimposed on the image above.
These phenomena are observable in all directions tangent to the sphere of photons: if spaceship searchlights beam in one of these directions, the observer will see in this direction the distorted image of the part of the spaceship which is behind him.

Observer located on the outside of the event horizon of the black hole (radial coordinate = \(R_s\))

All the observable stars in the universe (assumed to be located at a very great distance from the observer) will appear « gathered » above the observer in a disk of apparent elevation \(L_{crit}\) or about \(21.1^\circ\) \(\left(\arctan\left(\frac{2}{3\sqrt{3}}\right)\right)\), see paragraph 3.3.6. above), with small « contraction » at high elevations (no contraction at the elevation \(90^\circ\)), becoming more pronounced for low elevations and tending towards infinity for the limit \(L_{crit}\) of \(21.1^\circ\), the photons have to make several revolutions around the black hole to approach this limit and enter the event horizon). The edge of the disc will be made up of images of stars located precisely at the elevation \(90^\circ\) or \(-90^\circ\) (see above). The apparent diameters of stars are smaller than their real diameters, and decrease more and more significantly with the elevation. See general relativity appendix B.2 for more details.

Photon emitted with \(r\) emission \(<r\) critical and subjected to the gravitational field of a black hole

Note: the numerical values in meters calculated in this paragraph refer to a hypothetical black hole with the mass of the sun.

Emission radial coordinate \(<R_s\)

If the photon is emitted within the event horizon, it cannot approach it whatever its direction of emission¹¹, and will necessarily reach the center \(O\) of the black hole.

Emission radial coordinate \(=R_s\)

A photon \(p\) coming from a light source located on the outside of the event horizon can escape it only if \(b<b_{crit}\) (see paragraph 3.5.3 above), which corresponds to an angle of emission (angle to the plane tangent to the sphere with radius \(R_s\) at point of emission) greater than \(L_{crit}\) or about \(21.1^\circ\) \(\left(\arctan\left(\frac{2}{3\sqrt{3}}\right)\right)\).
In Schwarzschild spacetime, as a geodesic can be followed in either direction, this case corresponds to the one seen in paragraph 3.3.6 above (photon coming from infinity with \(b<b_{crit}\)).
See figure 6 above for the plot of the trajectory (\(b\) calculated for a \(\varphi\) variation of \(\pi\)).
If \(b\) is very slightly lower than \(b_{crit}\), \(p\) will make several revolutions around the black hole before escaping to infinity.
For \(b=b_{crit}\) (angle of emission \(=L_{crit}\)), \(p\) will move to the unstable orbit seen above (\(r=r_{crit}\)).
See figure 10 above for the plot of the trajectory.
For \(b>b_{crit}\), (angle of emission \(<L_{crit}\)), \(p\) will initially be exiting, then entering in a second phase, and it will return to the event horizon, its angle of impact being the same as its angle of emission in absolute terms \(\tan\ L_{emission}=\frac{R_s}{b}\) and \(\tan L_{reception}=-\frac{R_s}{b}\).
With, for example \(b=8\ 075\ m\) (calculated for a \(\varphi\) variation of \(\pi\) or ½ revolution), (3.k) gives the value of the apocentre: \(r_{apo}=3\ 784\ m\).
See figure 8 for the plot of the trajectory.
If \(b\) is very slightly greater than \(b_{crit}\), \(p\) will make several revolutions around the black hole before entering the event horizon.
See figure 9 for the plot of the trajectory (\(b\) calculated for a \(\varphi\) variation of \(4\pi\) or 2 revolutions).

Emission radial coordinate \(r_{em}\in[R_s,\ r\) critical \(]\)

The trajectory of \(p\) will be as described in the previous paragraph, depending on the value of \(b\) with respect to \(b_{crit}\).

Emission radial coordinate \(r_{em}=r\) critical

\(b\) cannot exceed \(b_{crit}\) which is \(b_{max}\) in this particular case (see paragraph 3.4 above, by writing \(r_{em}=r_{crit}\) in (3.s)).
For \(b=b_{crit}\), the angle of emission of \(p\) is zero resulting in \({du\over d\varphi}=0\) and \(p\) is in the unstable orbit \(r=r_{crit}\).
For \(b<b_{crit}\) , \(p\) is exiting to infinity if \({d\varphi\over dt}<0\) while \(p\) is entering the event horizon if \({d\varphi\over dt}>0\).

Image of black hole accretion disks

By definition, it is not possible to see a black hole. However, in the case of a stellar black hole with accretion disks, the light emitted by these disks will be deflected according to the rules seen above.

CONCLUSION

Applying classical mechanics for calculating the photon trajectories deflected by a massive object in spherical symmetry gives a significantly different result from that given by general relativity with the Schwarzschild metric.
With \(M\) mass of the object, \(R\) its radius, \(G\) gravitational constant, \(c\) speed of light in a vacuum, \(r\) radial coordinate of the photon, \(b\) impact parameter, \(R_s\) Schwarzschild radius \(=\frac{2GM}{c^2}\), \(e\) eccentricity \(=\sqrt{1+\frac {4b^2}{R_s^2}}\), \(bcrit=\frac{3\sqrt{3}}{2}R_s\) and \(L_{crit}=\arctan\left(\frac{2}{3\sqrt{3}}\right)\), the table below summarizes the main differences:

	Classical mechanics	General relativity	comments
Trajectory	analytical function (branch of hyperbola, parabola , ellipse or circle)	numerical integration
Pericentre (photon coming from infinity)	\(r_{per}=\frac{2b^2}{R_s(1+e)}\)	\(r_{per}=\frac{2b}{\sqrt{3}}\cos\left({1\over 3}\arccos\left(-\frac{b_{crit}}{b}\right)\right)\) with \(b>b_{crit}\) see figures 3 and 4
Minimum non-impact radial coordinate (photon coming from infinity)	\(R\)	\(\max\left(R,\frac{3}{2}R_s\right)\)	in general relativity, a photon coming from infinity cannot « approach » closer than\(\frac{3}{2}R_s\) without impacting the massive object.
Total deflection (photon coming from infinity)	\(\simeq\frac{R_s}{b}\)	\(\simeq\frac{2Rs}{b}\)
Maximum total deflection (photon coming from infinity)	\(180^\circ\)	no maximum value	in general relativity, photons can be captured in an unstable circular orbit around the massive object, creating a sphere of photons.
Apparent radius of a massive object with radius \(R\) between \(R_s\) and \(\frac{3}{2}R_s\)	\(\simeq R\)	\(b_{crit}\)
Capture (photon coming from infinity)	no	yes, on an orbit with radial coordinate \(\frac{3}{2}R_s\) with \(b=b_{crit}\) see figure 5	in general relativity, a photon can be captured in an unstable orbit around a massive object.
Impact (photon coming from infinity)	\(b<b_{lim}\) \(=R\sqrt{1+\frac{R_s}{R}}\)	if \(R>\frac{3}{2}R_s\) : \(b<b_{max}=R\sqrt{1-\frac{R_s}{R}}\) if \(R\le\frac{3}{2}R_s\) : \(b<b_{crit}\) with in the case \(R=R_s\) : \(L_{impact}>L_{crit}\simeq 21.1^\circ\) see figure 6
Stars visible from the event horizon of a black hole	all visible stars located between elevation \(-19.5^\circ\) and \(+90^\circ\), with an apparent diameter that decreases with the elevation	all visible stars in the universe, with an apparent diameter that decreases significantly with the elevation	in general relativity, all visible stars in the universe, from the event horizon of a black hole, appear « gathered » in a disk of elevation \(L_{crit}\simeq 21.1^\circ\).
Observation and shape of the image of a star located « behind » a black hole on the axis joining the observer and the center of the black hole	condition: the observer must be at a sufficient distance from the black hole. The image of the star is then a very thin circle (« Newton circle© »)	no condition on the distance between the observer and the event horizon of the black hole. The image of the star is a very thin circle (known as « Einstein ring »)	in general relativity, the diameter of the circle is greater than that calculated in classical mechanics.
Apocentre (photon emitted from \(r_{em}\) between \(R_s\) and \(\frac{3}{2}R_s\))	no	\(r_{apo}=\) \(\frac{2b}{\sqrt{3}}\cos\left({1\over 3}\arccos\left(-\frac{b_{crit}}{b}\right)+\frac{4\pi}{3}\right)\) with \(b>b_{crit}\) see figures 8 and 9	in general relativity, the photon can reach an apocentre and return to the event horizon.
Capture (photon emitted from \(r_{em}\) between \(\frac{R_s}{2}\) and \(\frac{3}{2}R_s\))	yes, with \(r_{em}=\frac{R_s}{2}\), on an orbit with radial coordinate = \(\frac{R_s}{2}\)	yes, with \(r_{em}\) between \(R_s\) and \(\frac{3}{2}R_s\)), on an orbit with radial coordinate \(\frac{3}{2}R_s\) reached with \(b=b_{crit}\) see figure 10	in general relativity, the photon can join the sphere of photons.
Release (photon emitted from \(r_{em}=R_s\))	no condition	\(b<b_{crit}\) or an angle of emission \(>L_{crit}\simeq 21.1^\circ\) see figure 6	the release trajectory of a photon corresponds to its impact trajectory.
Invariance of the speed of light in vacuum	no, in general case If entering from infinity: \(v(r)=c\sqrt{1+\frac {R_s}{r}}\) If exiting to infinity with \(r_{em}=R_s\) : \(v(r)=c\sqrt{1-\frac {R_s}{r}\left(1-\frac {R_s}{r}\right)}\) yes, in a specific case (capture on circular orbit \(r=\frac{R_s}{2}\))	yes	in classical mechanics, the speed of a photon in a gravitational field can be \(<\) or \(>c\)${\displaystyle >c}$.
Homothety of trajectories	yes	yes	in classical mechanics and general relativity, the trajectories of the photons are homothetic in the ratio\(\frac{M_2}{M_1}\) if \(b_2=\frac{M_2}{M_1}b_1\).

In conclusion, it appears that classical mechanics cannot calculate the photon trajectories deflected by a massive object in spherical symmetry. Within the framework of general relativity, the Schwarzschild metric and coordinates are a first approach, with limitations for:
– the study of astrophysical cases, as the large majority of objects are rotating and therefore non-spherical,
– the study of black holes themselves, the Schwarzschild radius being an immaterial barrier linked to the coordinate system used.
While remaining within the framework of general relativity, it is possible to overcome these limits with the Kerr metric and the Eddington-Finkelstein 3+1 coordinate system.
To sum up, general relativity currently provides the best explanation of the phenomena observed in the deflection of light by massive objects, and has highlighted other phenomena such as the red shift in the frequency of starlight for a terrestrial observer, the Shapiro effect (delay of light), which can be used to estimate the mass of celestial bodies located at very great distances from the solar system, or the clock shift of satellites, which must be corrected to achieve GPS accuracy.

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