Contents
- 1 Calculating \(L\) and \(\frac{dL}{d\varphi}\)
- 2 Solving \(r^3-rb^2+b^2R_s = 0\) by Cardan’s method and discussing the null geodesics in Schwarzschild’s spacetime
- 3 Table summarizing the various possible null geodesics in Schwarzschild’s spacetime
Calculating \(L\) and \(\frac{dL}{d\varphi}\)
Replacing \(du\over d\varphi\) by its value (3.j) in the expression \(\tan L=\frac {1}{u}{du \over d\varphi}\), we get:
\(\tan L=\pm\sqrt{R_su-1+\frac{1}{b^2u^2}}=\pm\sqrt{\frac{R_s}{r}-1+\frac{r^2}{b^2}}\hspace{2cm}\)(3.x),
with \({dL \over d\varphi}=\frac{d\tan L(\varphi)\over d\varphi}{1+\tan^2 L(\varphi)}=\frac{d\left({1\over u}\frac{du}{d\varphi}\right)\over d\varphi}{1+\tan^2 L(\varphi)}\), then:
\({dL\over d\varphi}=\frac{\frac{1}{u}{d^2u \over d\varphi^2}-\frac {1}{u^2}\left({du\over d\varphi}\right)^2}{\frac{R_s}{b^2u}+\frac{1}{b^2u^2}}\),
so by replacing \(d^2u\over d\varphi^2\) and \(\left({du\over d\varphi}\right)^2\) by their respective values given by (3.w) and (3.v):
\({dL\over d\varphi}=\frac{\frac{1}{u}\left({3\over 2}R_su^2-u\right)-{1\over u^2}(R_su^3-u^2+{1\over b^2})}{\frac{R_s}{b^2u}+\frac{1}{b^2u^2}}=\frac{\frac{R_su}{2}-\frac{1}{b^2u^2}}{R_su+\frac{1}{b^2u^2}}=\frac{\frac{R_s}{2r}-\frac{r^2}{b^2}}{\frac{R_s}{r}+\frac{r^2}{b^2}}\hspace{2cm}\)(3.y),
applicable to null geodesics in Schwarzschild’s spacetime.
Solving \(r^3-rb^2+b^2R_s = 0\) by Cardan’s method and discussing the null geodesics in Schwarzschild’s spacetime
(3.k) is equivalent to \(F(r)=r^3−rb^2+b^2R_s=0\hspace{2cm}\)(3.z),
noting that \(F(r)\) is always positive or zero because with (3.h) and after calculation we get:
\(F(r)={b\over r}\left({dr \over d\varphi}\right)^2\)
(3.z) is a depreciated polynomial equation of degree 3 in \(r\) which has three solutions \(r_0,\ r_1\) et \(r_2\) and can be solved by Cardan’s method1.
Setting \(F(r)=r^3+Ar+B=0\), the discriminant of the equation is:
\(\Delta=-(4A^3+27B^2)\) or with \(A=−b^2\) et \(B=b^2R_s\) :
\(\Delta =b^4(4b^2-27R_s^2)\).
If \(\Delta >0\), there are three distinct real solutions \(r_0,\ r_1\) et \(r_2\), if \(\Delta =0\), all three solutions are real and one is double and if \(\Delta <0\), only one solution is real and the other two solutions are complex conjugates2.
On the other hand, the solutions to be considered for (3.z) must belong to \([0,+\infty[\) (by the definition of \(r\) radial coordinate).
(3.z) being issued from (3.k), which assumes that \(b\) is non-zero, the positive value that cancels out \(\Delta\) is the critical value \(b_{crit}=\frac{3\sqrt{3}}{2}R_s\ (=\sqrt{3}\ r_{crit}\) with \(r_{crit}={3\over 2}R_s\)).
For the discussion below of the null geodesics in Schwarzschild’s spacetime, we consider a massive object of mass \(M\) represented by a sphere of radius \(R\) (the associated Schwarzschild’s radius being \(R_s=\frac{2GM}{c^2}\)).
Case \(b>b_{crit}\) or \(\Delta>0\)
All three solutions are real and distinct and can be written as:
\(r_k=2\sqrt{-{A\over 3}}\cos\left({1\over 3}\arccos\left(\frac{3B}{2A}{\sqrt{-3\over A}}\right)+2k{\pi\over 3}\right)\) with \(k\in (0,1,2)\)3,
and replacing A and B by their respective values leads to:
\(r_k={2b\over\sqrt{3}}\cos\left({1\over 3}\arccos\left(-\frac{b_{crit}}{b}\right)+2k{\pi\over 3}\right)\) with \(k\in (0,1,2)\).
Since \(b>b_{crit}\), \(-\frac{b_{crit}}{b}\in \ ]-1,0[\) therefore \(\arccos\left(-\frac{b_{crit}}{b}\right)\in ]{\pi\over 2},\pi[\) resulting in:
\({1\over 3}\arccos\left(-\frac{b_{crit}}{b}\right)\in ]{\pi\over 6},{\pi\over 3}[\) or \(r_0\in ]\frac{b}{\sqrt {3}},b[\),
\({1\over 3}\arccos\left(-\frac{b_{crit}}{b}\right)+{2\pi\over 3}\in ]{5\pi\over 6},\pi[\) or \(r_1\in ]-b,-\frac{2b}{\sqrt {3}}[\),
\({1\over 3}\arccos\left(-\frac{b_{crit}}{b}\right)+{4\pi\over 3}\in ]{3\pi\over 2},{5\pi\over 3}[\) or \(r_2\in ]0,\frac{b}{\sqrt {3}}[\).
The above ranges lead to \(r_1<0<r_2<r_0\)
\(\Rightarrow\) (3.z) has two solutions \(r_2\) et \(r_0\) on \(]0,+\infty[\).
For positive values of \(r\), \(F(r)\) has a positive or zero value on \(]0,r_2]\) and on \([r_0,\infty[\).
Photon entering from infinity
The radial coordinate \(r\) of \(p\) belongs to \([r_0,+\infty[\).
Assuming \(R<r_0\), \(r\) reaches \(r_0\) (passing pericentre \(F(r_0)=0\)) then \(p\) continues to infinity. If \(R>r_0\), \(p\) impacts the massive object when \(r=R\).
Note that \(r\) cannot reach \(r_2\) since \(r_2<r_0\).
Photon exiting the event horizon
The radial coordinate \(r\) of \(p\) belongs to \(]R_s,r_2[\).
\(r\) reaches \(r_2\) (passing apocentre \(F(r_2)=0\)) then \(p\) returns to the event horizon of the massive object.
The value of \(r\) cannot exceed \(r_2\) as this would lead to \(F(r)< 0\) in contradiction with \(F(r)\) positive or zero as previously stated.
The condition for a photon emitted from the event horizon of a black hole \(r_{em}=R_s\) to return to the event horizon is therefore \(b>b_{crit}\).
Case \(b=b_{crit}\) or \(\Delta=0\)
All three solutions are real and one is double:
\(r_1=\frac{3B}{A}\) et \(r_0=r_2=-\frac{3B}{2A}\)4,
replacing A and B by their respective values leads to:
\(r_1=-3R_s\) and \(r_0=r_2={3\over 2}R_s\)
\(\Rightarrow\) (3.z) has a solution \(r_{crit}={3\over 2}R_s\) on \(]0,+\infty[\).
Photon entering from infinity
Assuming that \(R<{3\over 2}R_s\), this is the limiting case where the radial coordinate \(r\) of \(p\) entering from infinity asymptotically reaches « from above » the value \(r_0=r_2=r_{crit}={3\over 2}R_s\) and \(p\) moves to a circular orbit of radius \(r_{crit}\) around the massive object. This orbit is unstable: as seen previously, if \(r\) becomes greater than \(r_{crit}=r_0\), \(p\) continues towards infinity, and if \(r\) becomes less than \(r_{crit}=r_2\), \(p\) follows a trajectory towards the point \(O\) center of the massive object and reaches the latter. If \(R>{3\over 2}R_s\), \(p\) impacts the massive object when \(r= R\).
Photon exiting the event horizon
Assuming that \(R<{3\over 2}R_s\), this is the limiting case where the radial coordinate \(r\) of \(p\) exiting from \(R_s\) asymptotically reaches « from below » the value \(r_2=r_0=r_{crit}={3\over 2}R_s\) and \(p\) moves to a circular orbit of radius \(r_{crit}\) around the massive object. This orbit is unstable: as seen previously, if \(r\) becomes less than \(r_{crit}=r_2\), \(p\) returns to the massive object, and if \(r\) becomes greater than \(r_{crit}=r_0\), \(p\) escapes from the massive object and follows a trajectory towards infinity.
Note that the massive object is not necessarily a black hole, since a compact object of radius \(R\in ]R_s,{3\over 2}R_s[\) allows the circular orbit of \(p\).
Case \(b<b_{crit}\) or \(\Delta<0\)
Only one solution is real and the other two are complex conjugates.5
The real solution is:
\(r_1=\sqrt[3]{\frac{-B+\sqrt{\frac{-\Delta}{27}}}{2}}+\sqrt[3]{\frac{-B-\sqrt{\frac{-\Delta}{27}}}{2}}\)6.
With \(b_{crit}^2=\frac{27R_s^2}{4}\), \(\Delta\) can be written:
\(\Delta=4b^4(b^2-b_{crit}^2)\),
and replacing B and \(\Delta\) by their respective values and after calculation:
\(r_1=\frac{\sqrt[3\over 2]{b}}{\sqrt{3}}(\sqrt[3]{-b_{crit}+\sqrt{b_{crit}^2-b^2}}-\sqrt[3]{b_{crit}+\sqrt{b_{crit}^2-b^2}})\).
Noting that \(-b_{crit}+\sqrt{b_{crit}^2-b^2}\) is \(<0\) (decreasing function of \(b\) and of zero value for \(b=0\)), it appears that \(r_1\) is \(<0\)
\(\Rightarrow\) (3.z) has no solution on \([0,+\infty[\).
Photon entering from infinity
The radial coordinate \(r\) of \(p\) is initially decreasing, which means that \(r\) having no minimum tends towards \(0\): \(p\) follows a trajectory towards the point \(O\) center of the massive object and hits the massive object, when \(r=R\) and whatever the value of \(R\).
\(L\) defined above represents for \(\varphi=\varphi_{impact}\) the angle of impact of \(p\) (angle with the tangent plane to the sphere of radius \(R\) at the point of impact).
\(b<b_{crit}\) leads to a condition on \(L_{\varphi_{impact}}\) according to (3.x):
\(\tan L_{\varphi_{impact}}\geq \sqrt{\frac{R_s}{R}-1+\frac {R^2}{b_{crit}^2}}\).
Contraction of angular directions
An observer located at the event horizon of the massive object at the point of impact would note an apparent direction \(\varphi_{apparent}=\frac{\pi}{2}-L_{\varphi_{impact}}\) for an actual direction of emission \(\simeq\varphi_{impact}\) (assuming \(R\ll\) distance of emission of the photon).
At the event horizon of a black hole, the contraction factor \(\frac{\varphi _{impact1}}{\varphi_{apparent}}\) is \(1\) for \(b=0\) and tends to infinity when \(b\) and \(L_{\varphi_{impact}}\) respectively tend towards \(b_{crit}\) and \(L_{crit}\) (the number of revolutions around the black hole and therefore \(\varphi_{impact}\) tend towards infinity).
Contraction of apparent star diameters
The contraction of the apparent diameters of the stars corresponds to the ratio \(\frac{Actual\ \Delta L}{Apparent\ \Delta L}\) for a specified elevation \(L\).
Numerical integration of the null geodesics in Schwarzschild’s spacetime leads to the following results for the event horizon of a black hole:
| \(Actual\ L\) | \(Apparent\ L\) | \(Contraction factor\) | \(\frac{Actual\ \Delta L}{Apparent\ \Delta L}\) |
| \(90^\circ\) | \(90^\circ\) | \(1\) | \(1\) |
| \(45^\circ\) | \(52.6^\circ\) | \(\simeq 1.20\) | \(\simeq 1.71\) |
| \(0^\circ\) | \(35.1^\circ\) | \(\simeq 1.64\) | \(\simeq 4.11\) |
| \(-45^\circ\) | \(27.6^\circ\) | \(\simeq 2.16\) | \(\simeq 9.18\) |
| \(-90^\circ\) | \(24.2^\circ\) | \(\simeq 2.73\) | \(\simeq 19.41\) |
| NA | \(\arctan\left(\frac{2}{3\sqrt {3}}\right) \simeq 21.1^{\circ }\) | \(\infty\) | \(\infty\) |
Photon exiting the event horizon
The radial coordinate \(p\) is initially increasing: since \(r\) has no maximum, \(p\) will escape from the massive object and follow a trajectory towards infinity, whatever the value of \(R\).
The condition for a photon emitted from the event horizon of a black hole (\(r_{em}=R_s\)) to escape is therefore \(b<b_{crit}\).
Table summarizing the various possible null geodesics in Schwarzschild’s spacetime
| Radial emission coordinate | Impact parameter | Sign of \(\frac{d\varphi}{dt}\) | Extremum of \(r\) | Overview |
| \(]r_{crit},+\infty[\) | \(]b_{crit},b_{max}]\) | \(+\) | pericentre \(r_{min}=r_{per}\) | photon entering from \(r_{em}\) then exiting to \(\infty\) |
| \(]r_{crit},+\infty[\) | \(b_{crit}\) | \(+\) | \(r_{min}=r_{crit}\) | photon entering from \(r_{em}\) then moving to the unstable orbit \(r=r_{crit}\) |
| \([R_s,r_{crit}[\) | \([b_{crit},b_{max}]\) | \(+\) | NA | photon entering from \(r_{em}\) in the event horizon |
| \([R_s,+\infty[\) | \([0,b_{crit}[\) | \(+\) | NA | photon entering from \(r_{em}\) in the event horizon |
| \([R_s,r_{crit}[\) | \(]b_{crit},b_{max}]\) | \(-\) | apocentre \(r_{max}=r_{apo}\) | photon exiting from \(r_{em}\) then entering in the event horizon |
| \([R_s,r_{crit}[\) | \(b_{crit}\) | \(-\) | \(r_{min}=r_{crit}\) | photon exiting from \(r_{em}\) then moving to the unstable orbit \(r=r_{crit}\) |
| \(]r_{crit},+\infty[\) | \([b_{crit},b_{max}]\) | \(-\) | NA | photon exiting from \(r_{em}\) to \(\infty\) |
| \([R_s,+\infty[\) | \([0,b_{crit}[\) | \(-\) | NA | photon exiting from \(r_{em}\) to \(\infty\) |
| \(r_{crit}\) | \(b_{crit}\) | \(+/-\) | \(r=r_{crit}\) | photon on the unstable orbit \(r=r_{crit}\) |
| \(r_{crit}\) | \(]b_{crit},+\infty[\) | NA | NA | case impossible as \(b>b_{max}=\frac{r_{crit}}{\sqrt {1-{\frac {Rs}{r_{crit}}}}}=b_{crit}\) |
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