Contents
- 1 Calculating \(L\) and \(\frac{dL}{d\phi}\)
- 2 Solving \(r^3-rb^2+b^2R_s = 0\) by Cardan method and discussing the null geodesics in Schwarzschild spacetime
- 3 Table summarizing the various possible null geodesics emitted at a radial coordinate \(r_{em}\) in Schwarzschild spacetime
Calculating \(L\) and \(\frac{dL}{d\phi}\)
The complementary angle \(V\) (tangent to the trajectory) is defined by \(\tan L=\frac {1}{u}{du \over d\phi}\)
giving with (3.j):
\(\tan L=\pm\sqrt{Rs_u-1+\frac{1}{b^2u^2}}=\pm\sqrt{\frac{R_s}{r}-1+\frac{r^2}{b^2}}\hspace{2cm}\)(3.x)
knowing that \({dL \over d\phi}=\frac{d\tan L(\phi)\over d\phi}{1+\tan^2 L(\phi)}=\frac{d({1\over u}\frac{du}{d\phi})\over d\phi}{1+\tan^2 L(\phi)}\), we get:
\({dL\over d\phi}=\frac{\frac{1}{u}{d^2u \over d\phi^2}-\frac {1}{u^2}({du\over d\phi})^2}{\frac{R_s}{b^2u}+\frac{1}{b^2u^2}}\)
Or by replacing \(d^2u\over d\phi^2\) and \(({du\over d\phi})^2\) by their respective values given by (3.v) et (3.j):
\({dL\over d\phi}=\frac{\frac{1}{u}({3\over 2}R_su^2-u)-{1\over u^2}(R_su^3-u^2+{1\over b^2})}{\frac{R_s}{b^2u}+\frac{1}{b^2u^2}}=\frac{\frac{R_su}{2}-\frac{1}{b^2u^2}}{R_su+\frac{1}{b^2u^2}}=\frac{\frac{R_s}{2r}-\frac{r^2}{b^2}}{\frac{R_s}{r}+\frac{r^2}{b^2}}\hspace{2cm}\)(3.y)
applicable to null geodesics in Schwarzschild spacetime.
Solving \(r^3-rb^2+b^2R_s = 0\) by Cardan method and discussing the null geodesics in Schwarzschild 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:
\(F(r)={b\over r}({dr \over d\phi})^2\)
(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).
Equation (3.z) being issued from (3.k), which assumes that \(b\) is non-zero, the positive value that nullifies \(\Delta\) 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 spacetime, we consider a massive object of mass \(M\) represented by a sphere of radius \(R\) (the associated Schwarzschild radius being \(R_s=\frac{2GM}{c^2}\)).
Case \(b>b_{crit}\) or \(\Delta>0\)
All three solutions are real and can be written as:
\(r_k=2\sqrt{-{A\over 3}}\cos({1\over 3}\arccos(\frac{3B}{2A}{\sqrt{-3\over A}})+2k{\pi\over 3})\) with \(k\in (0,1,2)\)3
and replacing A and B by their respective values leads to:
\(r_k={2b\over\sqrt{3}}\cos({1\over 3}\arccos(-\frac{b_{crit}}{b})+2k{\pi\over 3})\) with \(k\in (0,1,2)\)
Since \(b>b_{crit}\), \(-\frac{b_{crit}}{b}\in \ ]-1,0[\) therefore \(\arccos(-\frac{b_{crit}}{b})\in ]{\pi\over 2},\pi[\) resulting in:
\({1\over 3}\arccos(-\frac{b_{crit}}{b})\in ]{\pi\over 6},{\pi\over 3}[\) or \(r_0\in ]\frac{b}{\sqrt {3}},b[\),
\({1\over 3}\arccos(-\frac{b_{crit}}{b})+{2\pi\over 3}\in ]{5\pi\over 6},\pi[\) or \(r_1\in ]-b,-\frac{2b}{\sqrt {3}}[\),
\({1\over 3}\arccos(-\frac{b_{crit}}{b})+{4\pi\over 3}\in ]{3\pi\over 2},{5\pi\over 3}[\) 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 coming 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 \(r_2<r_0\).
Note: a photon \(p\) with radial coordinate \(<r_2\) will head for the point \(O\) center of the massive object and hit the latter (the value of \(r\) cannot exceed \(r_2\) as this would lead to \(F(r)< 0\) which is impossible since \(F(r)\) is positive or zero as previously stated).
Photon leaving the event horizon of the massive object towards the outer space
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.
Note: as seen previously, the value of \(r\) cannot exceed \(r_2\).
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 coming 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 leaving the event horizon of the massive object towards the outer space
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 \(p\) described above.
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
By replacing in (3.s) \(b_{crit}\) by its value, \(\Delta\) 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 coming 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.w):
\(\tan L_{\varphi_{impact}}\geq \sqrt{\frac{R_s}{R}-1+\frac {R^2}{b_{crit}^2}}\)
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 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 | \(\arctan(\frac{2}{3\sqrt {3}}) \simeq 21.1^{\circ }\) | \(\infty\) | \(\infty\) |
Photon leaving the event horizon of the massive object towards the outer space
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 (\(r=R_s\)) to escape is therefore \(b<b_{crit}\).
Table summarizing the various possible null geodesics emitted at a radial coordinate \(r_{em}\) in Schwarzschild 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 |
| \([R_s,r_{crit}[\) | \(]b_{crit},b_{max}]\) | \(-\) | apoastre \(r_{max}=r_{apo}\) | 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 \(b>b_{max}=\frac{r_{crit}}{\sqrt {1-{\frac {Rs}{r_{crit}}}}}=b_{crit}\) |
- https://www.techno-science.net/glossaire-definition/Methode-de-Cardan.html ↩︎
- https://www.techno-science.net/glossaire-definition/Methode-de-Cardan.html ↩︎
- https://www.techno-science.net/glossaire-definition/Methode-de-Cardan.html ↩︎
- https://www.techno-science.net/glossaire-definition/Methode-de-Cardan.html ↩︎
- https://www.techno-science.net/glossaire-definition/Methode-de-Cardan.html ↩︎
- https://www.techno-science.net/glossaire-definition/Methode-de-Cardan.html ↩︎