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TRAJECTORIES AND DEFLECTIONS OF LIGHT IN A GRAVITATIONAL FIELD GENERATED BY A MASSIVE OBJECT

1. INTRODUCTION

The deflection of light in a gravitational field is a phenomenon predicted by classical mechanics and by general relativity.

The two theories, however, give significantly different results, general relativity notably predicting unusual phenomena which depend on the locations of the observer and of the light source.

2. CLASSICAL MECHANICS

Warning: the speed of a photon is by definition relativistic, thus classical mechanics does not explain the observed results.

According to classical mechanics, the orbit of the photon is planar, and its trajectory is a branch of a hyperbola whose focus is the center of gravity of the massive object.

Polar coordinates (r, φ) and impact parameter b

Its equation can be written, with the inverse radial coordinate $u={\frac {1}{r}}$:

$u(\varphi )={\frac {Rs}{2b^{2}}}(1+e\cos(\varphi -\varphi _{0}))$ with:

$Rs={\frac {2GM}{c^{2}}}$ ($G$ 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 = ${\sqrt {1+{\frac {4b^{2}}{Rs^{2}}}}}$ and $\varphi _{0}$ symmetry axis = $\arccos(-{\frac {1}{e}})$.

For a given massive object, the trajectory of the photon is fully determined by the impact parameter $b$.

Photon coming with c speed from infinity

The speed $v(r)$ of the photon is $c{\sqrt {1+{\frac {Rs}{r}}}}$, 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 $b>R{\sqrt {1+{\frac {Rs}{R}}}}$, the photon does not impact the massive object, its minimum distance (at the pericentre) is ${\frac {2b^{2}}{Rs(1+e)}}$ , and the photon continues towards infinity on a trajectory 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 $2\arccos {\biggl (}-{\frac {1}{1+{\frac {2R}{Rs}}}}{\biggr )}-\pi$ which gives for the sun ($R\odot$ $6.96342\ 10^{8}m$, $M\odot$ $1.9891\ 10^{30}kg$), and with $G$ gravitational constant $6.6743\ 10^{-11}m^{3}.kg^{-1}.s^{-2}$ and $c$ 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 $Rs$ a value of $2\arccos {\bigl (}-{\frac {1}{3}}{\bigr )}-\pi$ or approximately $39^{\circ }$.

If $2b\gg Rs$, the total deflection is $\simeq {\frac {Rs}{b}}{\bigl (}={\frac {2GM}{c^{2}b}}{\bigr )}$.

With $b<R{\sqrt {1+{\frac {Rs}{R}}}}$, the photon impacts the massive object.

An observer standing on the surface of a massive object of radius $Rs$ and looking at the sky above him, would see observable stars whose true latitude lies between $90^{\circ }$ et $90^{\circ }-\arccos {\bigl (}-{\frac {1}{3}}{\bigr )}$ ( $\simeq -19.5^{\circ }$), the "contraction" being slight for latitudes close to $90^{\circ }$, and a little more pronounced for apparent latitudes 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 ${\frac {4}{3}}$ for apparent latitude $0^{\circ }$.

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 $c{\sqrt {1-{\frac {Rs}{R}}{\bigl (}1-{\frac {R}{r}}{\bigr )}}}$, which is zero at $\infty$ if $R=R_{s}$.

3. GENERAL RELATIVITY

The elementary displacement of the photon is a like-light vector and its scalar product is zero.

Assuming that the gravitational field is spherically symmetrical, and applying the Schwarzschild metric (see its limits in the conclusion), the orbit of the photon remains in a plane ($\theta =cte$), and the scalar product of the elementary displacement $(cdt,dr,d\theta ,d\varphi )$ can be written with $d\theta =0$:

$-{\bigl (}1-{\frac {Rs}{r}}{\bigr )}c^{2}dt^{2}+{\biggl (}{\frac {1}{1-{\frac {Rs}{r}}}}{\biggr )}dr^{2}+r^{2}d\varphi ^{2}=0$.

Note: in the asymptotic region $r\gg Rs$, the coordinate $r$ is interpreted as the physical distance between the photon and the center of the massive object.

The previous equation and the conservation of energy and angular momentum of the photon along its geodesic give the trajectory of the photon, by integrating

${du \over d\varphi }=\pm {\sqrt {Rs\ u^{3}-u^{2}+{\frac {1}{b^{2}}}}}$

with the $u$ inverse radial coordinate $={\frac {1}{r}}$, its initial value and with:

$Rs=$ 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 $\varphi =0$).

For a given massive object, the photon trajectory is fully determined by the impact parameter $b$ and has different shapes depending on the value of $b$ with respect to $b_{crit}={\frac {3{\sqrt {3}}}{2}}Rs$ which cancels the discriminant of ${\bigl (}{du \over d\varphi }{\bigr )}^{2}$.

Fig. A - 4 trajectories of a photon coming from $\infty$ around a Schwarzschild black hole (deflections ${\frac {\pi }{2}},\ \pi ,{\frac {3\pi }{2}}\ and\ 2\pi$)

Photon coming from infinity

1) If $b>b_{crit}$, and under the condition that the photon does not impact the massive object, assumed to be spherical with radius $R$, that is $b>b_{lim}=R{\sqrt {\frac {1}{1-{\frac {Rs}{R}}}}}$, its radial coordinate $r$ decreases to its minimum (at the pericentre, cancellation of ${du \over d\varphi }$) which is $r_{per}={\frac {2b}{\sqrt {3}}}\cos {\bigl (}{\frac {1}{3}}\arccos \ {\bigl (}-{\frac {b}{b_{crit}}}{\bigr )}{\bigr )}$, 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 $2\textstyle \int _{r_{per}}^{\infty }{\frac {1}{r^{2}{\sqrt {{\frac {1}{b^{2}}}-{\frac {1}{r^{2}}}({1-{\frac {Rs}{R}}})}}}}dr-\pi$.

Fig. B - Trajectory of a photon coming from $\infty$ and circling twice a Schwarzschild black hole

If the radius $R$ is $\gg Rs$ since $b>R$, $b$ is $\gg Rs$ and a limited development in ${\frac {Rs}{b}}$ provides a total deflection

$\simeq {\frac {2Rs}{b}}{\bigl (}={\frac {4GM}{c^{2}b}}{\bigr )}$, which gives with $b=b_{lim}$ 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 precision of measurement, the photographs of the solar disk vicinity taken by Arthur Eddington and his team during the total eclipse on the island of Principe on May 29 1919 confirmed this value (which is twice the value of the classical mechanics theory calculated above).

Fig. C - Trajectory of a photon coming from $\infty$ and caught by a Schwarzschild black hole

Fig. D - Trajectory of a photon coming from $\infty$ and absorbed by a Schwarzschild black hole or a photon emitted towards $\infty$ from this black hole

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 $\pi$.

2) If $b=b_{crit}$, ${du \over d\varphi }$ becomes zero for $r_{crit}={\frac {3}{2}}Rs$ and if $R<{\frac {3}{2}}Rs$, 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 sphere of photons with radial coordinate $r_{crit}={\frac {3}{2}}Rs\ {\bigl (}={\frac {3GM}{c^{2}}}{\bigr )}$ coming from stars with impact parameter $b_{crit}$.

Note: this sphere cannot be seen as such and reduces for an observer placed at the radial coordinate ${\frac {3}{2}}Rs$ to a very thin luminous ring at latitude $0^{\circ }$ of the observer.

3) If $b<b_{crit}$, $r$ 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 $Rs$) and looking at the sky above him, would see all the observable stars in the universe gathered in a disk of apparent latitude $L_{crit}=\arctan {\bigl (}{\frac {2}{3{\sqrt {3}}}}{\bigr )}$ that is $\simeq 21^{\circ }$, the "contraction" being minor for high latitudes (no contraction for $90^{\circ }$ apparent), becoming more pronounced for lower latitudes 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. The apparent diameters of stars are smaller than their real diameters, and decrease more and more significantly with latitude.

A star located precisely at latitude $-90^{\circ }$, 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 a latitude close to $L_{crit}$.

Fig. E - Trajectory of a photon emitted from a Schwarzschild black hole and back to it

Fig. F - Trajectory of a photon emitted from a Schwarzschild black hole and caught by it

Photon emitted from the radial coordinate Rs

1) If $b>b_{crit}$, the radial coordinate $r$ of the photon increases until it reaches its maximum value (at apocentre, cancellation of ${du \over d\varphi }$) which is$r_{apo}={\frac {2b}{\sqrt {3}}}\cos {\bigl (}{\frac {1}{3}}\arccos {\bigl (}-{\frac {b}{b_{crit}}}{\bigr )}\ +{\frac {4\pi }{3}}{\bigr )}$ 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.

2) If $b=b_{crit}$, 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).

3) If $b<b_{crit}$, $r$ 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.

A photon emitted from the event horizon of a black hole ($r=Rs$) can then be released from it under the condition $b<b_{crit}$ or an emission latitude $>L_{crit}$.

Note: for a same value of $b$, the geodesic of the photon is identical to that seen previously (arrival from infinity and impact on the massive object), but run in the opposite direction (figure D).

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.

Fig. G - Example of an apparent image of the accretion disks of a Schwarzschild black hole

The "hat" corresponds to photons passing "above" the black hole and the "hair and necklace" correspond to photons passing "below".

The apparent radius of the black hole is $b_{crit}={\frac {3{\sqrt {3}}}{2}}Rs$ since no photon of impact parameter $b<b_{crit}$ can reach the observer.

4. CONCLUSION

Classical mechanics cannot predict the behavior of photons in an intense gravitational field created by a massive object. Within 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 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 obtain GPS accuracy.

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