Contents
INTRODUCTION
The deflection of light in a gravitational field generated by a massive object is a phenomenon predicted by both classical mechanics and general relativity.
In the context of spherical symmetry, however, the deflection of light by black holes according to Newton or Schwarzschild gives significantly different results for the trajectories of the photons, with general relativity predicting unusual phenomena that depend on the locations of the observer and the light source.
CLASSICAL MECHANICS – LIGHT DEFLECTION ACCORDING TO NEWTON
Warning: as the photon has no mass, the results given by classical mechanics are not in line with the observations.
According to classical mechanics, the photon orbit is planar, and its trajectory is a conic whose focus is the center of gravity of the massive object.
Its equation can be written, with the inverse radial coordinate \(u={1\over r}\):
\(u(\varphi)=\frac {R_s} {2b^2}(1+e\cos(\varphi-\varphi_0))\)1, with:
\(R_s=\frac {2GM} {c^2}\), (\(G\) gravitational constant,
\(M\) mass of the massive object and \(c\) speed of light in vacuum),
\(b\) impact parameter (perpendicular distance between the path of the photon coming from infinity and the axis \(\varphi=0\)),
\(e\) eccentricity \(=\sqrt{1+\frac{4b^2}{R_s^2}}\) and \(\varphi_0\) symmetry axis \(=\arccos\left(-\frac{1}{e}\right)\).
For a given massive object, the trajectory of the photon is fully determined by the impact parameter \(b\) and this principle applies to the deflection of light by black holes according to Newton or Schwarzschild.
Photon coming with \(c\) speed from infinity
The speed \(v(r)\) of the photon is \(c\sqrt{1+\frac{R_s}{r}}\), which shows that in classical mechanics, the speed of light is not an invariant.
For a massive object assumed to be spherical with a radius \(R\), if \(b > R\sqrt{1+\frac{R_s}{R}}\) the photon does not impact the massive object, its minimum distance (at the pericentre) is \(\frac{2b^2}{R_s(1+e)}\), and the photon continues towards infinity on a branch of hyperbola symmetrical with respect to the axis \(\varphi_0\) with a total angular deflection of \(2\varphi_0-\pi\).
With the limit value of \(b\), the maximum deflection is \(2\arccos\left(-\frac{1}{1+\frac{2R}{R_s}}\right)-\pi\) which gives for the sun (\(R\odot 6.96342\ 10^8 m\), \(M\odot 1.9891\ 10^{30}kg\)), and with \(G\) gravitational constant \(6.6743\ 10^{-11}m^3.kg^{-1}.s^{-2}\) et \(c\) speed of light in vacuum \(299\ 792 \ 458\ m.s{-1}\), a value of \(0.875\) second of arc, and for a massive object with the mass of the sun and radius \(R_s\) a value of \(2\arccos\left(-{1\over3}\right)\ -\ \pi\) or approximately \(39^\circ\).
If \(2b \gg R_s\), the total deflection is \(\simeq {R_s\over b}\left(=\frac{2GM}{c^2b}\right)\).
With \(b < R\sqrt{1+\frac{R_s}{R}}\), the photon impacts the massive object.
An observer standing on the surface of a massive object of radius \(R_s\) and looking at the sky above him, would see observable stars whose true elevation lies between \(90^\circ\) and \(90^\circ- \arccos(-{1\over3})\) (\(\simeq -19.5^\circ\)), the « contraction » being slight for elevations close to \(90^\circ\), and a little more pronounced for apparent elevations close to \(0^\circ\). This phenomenon also applies to the apparent diameters of the stars, which are smaller than the real diameters with a maximum contraction factor of \(4\over 3\) for the apparent elevation \(0^\circ\).
Photon emitted from the surface of a massive spherical object of radius \(R\)
If \(c\) is the speed of the photon at emission, its speed \(v(r)\) is \(c\sqrt{1-\frac{R_s}{R}\left(1-\frac{R}{r}\right)}\), which is zero at \(\infty\) if \(R=R_s\).
GENERAL RELATIVITY – LIGHT DEFLECTION ACCORDING TO SCHWARZSCHILD
An elementary displacement of the photon is a light-like vector and its scalar product is zero2 (hence the name « null geodesics » for the photon trajectories).
Assuming that the gravitational field is spherically symmetrical, and applying the Schwarzschild metric (see its limits in the conclusion), the photon orbit remains in a plane (\(\theta=\) const), and the scalar product of the elementary displacement \((cdt, dr, d\theta, d\varphi)\) can be written with \(d\theta=0\):
\(-\left(1-\frac{R_s}{r}\right)c^2dt^2+\left(\frac{1}{1-\frac{R_s}{r}}\right)dr^2+r^2d\varphi^2 = 0\)3.
The coefficients of the metric are independent of \(t\) and the components of the metric tensor matrix are diagonal: the geometry of Schwarzschild spacetime is therefore static, and by definition spherically symmetrical.
Note: in the asymptotic region \(r \gg R_s\), the coordinate \(r\) is interpreted as the physical distance between the photon and the center of the massive object.
The previous equation and the invariance of energy and angular momentum of the photon along its geodesic give the trajectory of the photon, by integrating \(\frac{du}{d\varphi}=\pm\sqrt{R_su^3-u^2+\frac{1}{b^2}}\)4
with \(u\) the inverse radial coordinate \(={1\over r}\) and with:
\(R_s =\) Schwarzschild radius \(=\frac{2GM}{c^2}\), (\(G\) gravitational constant, \(M\) mass of the massive object and \(c\) speed of light in vacuum),
\(b\) impact parameter (perpendicular distance between the path of the photon coming from infinity and the axis \(\varphi=0\), see figure above).
For a given massive object and the initial value of \(u\), the photon trajectory is fully determined by the impact parameter \(b\) and this principle applies to the deflection of light by black holes according to Newton or Schwarzschild. The trajectory has different shapes depending on the value of \(b\) with respect to a critical value \(b_{crit}=\frac{3\sqrt{3}}{2}R_s\) which cancels out the discriminant of \(\left({du\over {d\varphi}}\right)^2\)5.
Details of the RK 4 double integration method
Photon coming from infinity
Impact parameter \(b>b\) critical
Under the condition that the photon does not impact the massive object, assumed to be spherical with radius \(R\) that is \(b>b_{lim}=R\sqrt{\frac{1}{1-\frac{R_s}{R}}}\), its radial coordinate \(r\) decreases to its minimum (at the pericentre, cancellation of \(\frac{du}{d\varphi}\)) which is \(r_{per}=\frac{2b}{\sqrt{3}}\cos\left({1\over 3}\arccos\left(-\frac{b_{crit}}{b}\right)\right)\)6,
and the photon continues towards infinity on a trajectory symmetrical with respect to the axis \(\varphi_{per}\) (value of \(\varphi\) at the pericentre) with a total angular deflection of \(2\int_{r_{per}}^{\infty}\frac{1}{r^2\sqrt{\frac{1}{b^2}-\frac{1}{r^2}(1-\frac{R_s}{R})}}dr-\pi\).
If the radius \(R\) is \(\gg R_s\) since \(b>R\), \(b\) is \(\gg R_s\) and a limited expansion in \(R_s\over b\) provides a total deflection \(\simeq\frac{2R_s}{b}\left(=\frac{4GM}{c^2b}\right)\)7, which gives with \(b=b_{lim}\)
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for the sun (\(R\odot 6.96342\ 10^8 m\), \(M\odot 1.9891\ 10^{30}kg\)), with \(G\) gravitational constant \(6.6743\ 10^{-11}m^3.kg^{-1}.s^{-2}\) and \(c\) speed of light in vacuum \(299\ 792 \ 458\ m.s^{-1}\), a value of \(1.750\) second of arc.
Note: to the accuracy of measurement, the photographs of the solar 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 above).
If the massive object is a black hole, there is no maximum mathematical value for the deflection: for \(b\) very close to \(b_{crit}\), the photon may circle the black hole several times before continuing on to infinity. The maximum physical value of the deflection is therefore \(\pi\).
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Impact parameter \(b=b\) critical
\(du\over d\varphi\) becomes zero for a critical value \(r_{crit}={3\over 2}R_s\) and if \(R<{3\over 2}R_s\), the photon moves to an unstable circular orbit of radius \(r_{crit}\) around the massive object, which means that a photon cannot « tangent » a massive object of radius \(<r_{crit}\).
A massive object of radius \(<r_{crit}\) is therefore surrounded by a spherical surface of photons with radial coordinate \(r_{crit}={3\over 2}R_s\left(=\frac{3GM}{c^2}\right)\) coming for instance from stars or accretion disks, with impact parameter \(b_{crit}\).
Note: this sphere cannot be seen as such and reduces for an observer placed at the radial coordinate \({3\over 2}R_s\) to a very thin luminous ring at the elevation \(0^\circ\) of the observer.
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Impact parameter \(b<b\) critical
\(r\) has no minimum and the photon impacts the massive object, without any condition on the value of its radius \(R\).
An observer standing on the event horizon of a black hole (an immaterial « surface » with radial coordinate \(R_s\)) and looking at the sky above him, would see all the observable stars in the universe gathered in a disk of apparent elevation \(L_{crit}=\arctan\left(\frac{2}{3\sqrt{3}}\right)\) that is \(\simeq 21^\circ\), the « contraction » being minor for high elevations (no contraction for \(90^\circ\) apparent), becoming more pronounced for lower elevations and tending towards infinity for the limit \(L_{crit}\), photons circling several times the black hole to approach this limit and re-entering the event horizon.
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The apparent diameters of stars are smaller than their real diameters, and decrease more and more significantly with the elevation.
A star located precisely at the elevation \(-90^\circ\) that is « behind » the black hole on the axis passing through its center and through the observer, will appear to the latter as very thin luminous circles centered on this axis (« Einstein rings ») above an elevation close to \(L_{crit}\).
Photon emitted from the radial coordinate \(R_s\)
Impact parameter \(b<b\) critical
The radial coordinate \(r\) of the photon increases until it reaches its maximum value (at apocentre, cancellation of \(\frac{du}{d\varphi}\)) which is \(r_{apo}=\frac{2b}{\sqrt{3}}cos\left({1\over 3}\arccos\left(-\frac{b_{crit}}{b}\right)+\frac{4\pi}{3}\right)\)8 and the photon follows on a symmetrical trajectory with respect to the axis \(\varphi_{apo}\) (value of \(\varphi\) at apocentre) then returns in the event horizon.
Impact parameter \(b=b\) critical
The result is identical to that seen previously for \(b=b_{crit}\): the photon moves to an unstable circular orbit of radius \(r_{crit}\) around the massive object (sphere of photons).
Impact parameter \(b<b\) critical
\(r\) has no maximum and the photon moves away from the massive object towards infinity. For \(b\) sufficiently close to \(b_{crit}\), the photon may circle several times the black hole before escaping to infinity.
In Schwarzschild spacetime, as a geodesic can be followed in either direction, this case corresponds to that seen above (fig. D).
A photon emitted from the event horizon of a black hole (\(r=R_s\)) can then be released from it under the condition \(b<b_{crit}\) or an emission angle \(>L_{crit}\).
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Black hole appearance
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 follow the rules seen above and an example of an apparent image is given in figure G.
The « hat » corresponds to photons passing « above » the black hole and the « hair and necklace » correspond to photons passing « below ».
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The apparent radius of the black hole is \(b_{crit}=\frac{3\sqrt{3}}{2}R_s\) since no photon of impact parameter \(b<b_{crit}\) can reach an observer. The shadow of the black hole corresponds to the disk of radius \(b<b_{crit}\).
CONCLUSION
The deflection of light by black holes according to Newton or Schwarzschild shows, on the one hand, that classical mechanics cannot predict the behavior of photons in an intense gravitational field created by a massive object and, on the other hand, that in the framework of general relativity, the Schwarzschild metric and coordinates are a first step, but with limitations for:
– the study of astrophysical phenomena, since the vast majority of objects are rotating and therefore not spherical,
– the study of black holes themselves, the Schwarzschild radius being an immaterial barrier linked to the coordinate system used,
limits that can be overcome with the Kerr’s 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 black holes, and has highlighted other phenomena such as the red shift in the frequency of starlight for a terrestrial observer, the Shapiro’s 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 compensated to obtain GPS accuracy.
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- https://luth.obspm.fr/~luthier/gourgoulhon/fr/master/relatM2.pdf ↩︎
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