Dr. Gyula Bene Department for Theoretical Physics, Loránd Eötvös University Pázmány Péter sétány 1/A, 1117 Budapest 3. week
The laws of physics in curved spacetime
Motion of mass point in gravitational field
Principle of least action:
\[\begin{align}
\delta S=-mc \delta\;\int ds =0
\end{align}\]
Equation of motion:
\[\begin{align}
\frac{Du^i}{Ds}=0
\end{align}\]
( \(u^i=\frac{dx^i}{ds}\) stands for the four velocity), i.e.
\[\begin{align}
\frac{d^2 x^i}{ds^2}+\Gamma^i_{\phantom{1}kl}\frac{d x^k}{ds}\frac{d x^l}{ds}=0
\end{align}\]
Hamilton-Jacobi equation
\[\begin{align}
p^i=mcu^i
\end{align}\]
\[\begin{align}
p_ip^i=m^2c^2
\end{align}\]
\[\begin{align}
g^{ik}\frac{\partial S}{\partial x^i}\frac{\partial S}{\partial x^k}-m^2c^2=0
\end{align}\]
Light propagation
\[\begin{align}
\frac{d k^i}{d\lambda}+\Gamma^i_{\phantom{1}kl}k^k k^l=0
\end{align}\]
where the wave number vector \(k^i\) is defined as
\[\begin{align}
k^i=\frac{dx^i}{d\lambda}\;,
\end{align}\]
or, alternatively, as
\[\begin{align}
k_i=\frac{\partial \psi}{\partial x^i}\;,
\end{align}\]
\(\psi\) being the phase (also called eikonal).
Weak gravitational field
Non-relativistic Lagrangian:
\[\begin{align}
L=-mc^2+\frac{mv^2}{2}-m\varphi
\end{align}\]
Since the relativistic action is \(-mc\int ds\), we have
\[\begin{align}
-mc \;ds=(-mc^2+\frac{mv^2}{2}-m\varphi)dt\;,
\end{align}\]
or
\[\begin{align}
ds= c dt+\frac{\varphi}{c}dt-\frac{v^2}{2c}dt\;.
\end{align}\]
Taking the square, we have (up to \(\cal{O} (1)\) in \(1/c\) )
\[\begin{align}
ds^2=(c^2+2\varphi)dt^2-d\bf{r}^2\;,
\end{align}\]
since \(v^2dt^2=d\bf{r}^2\). Therefore,
\[\begin{align}
g_{00}=1+\frac{2\varphi}{c^2}
\end{align}\]
Static gravitational field, gravitational redshift
Change of frequency during light propagation in stationary gravitational field:
\[\begin{align}
\frac{d k_0}{d\lambda}-\Gamma_{l0m}k^l k^m=0
\end{align}\]
Remember the expression of Christoffel's symbols:
\[\begin{align}
\Gamma_{l0m}=\frac{1}{2}\left(g_{l0,m}+g_{lm,0}-g_{m0,l}\right)
\end{align}\]
Since the field is stationary, the middle term vanishes. The remaining
expression is antisymmetric in the indices \(l,m\), therefore
\(\Gamma_{l0m}k^l k^m=0\). This implies that in a stationary gravitational
field
\[\begin{align}
\frac{d k_0}{d\lambda}=0\;
\end{align}\]
i.e., the frequency \(k_0\) is unchanged during propagation.
Frequency measured in proper time:
\[\begin{align}
\omega=\frac{k_0}{\sqrt{g_{00}}}\approx k_0\left(1-\frac{\varphi}{c^2}\right)
\end{align}\]
Let \(\varphi_1,\varphi_2\) be the Newtonian gravitational potentials at
emission and absorbtion of a light ray, respectively. We then have the
frequency change (compared to the frequency of the source)
\[\begin{align}
\Delta \omega=\frac{\varphi_1-\varphi_2}{c^2}\omega\phantom{reds}
\end{align}\]
when observing it. The Newtonian potentials are negative. If the source
is placed near a heavier celestial body (say, near the Sun) than the Earth
where the detector is located, and hence \(|
\varphi_1|>|\varphi_2|\), we have \(\Delta \omega<0\), the gravitational redshift.
Maxwell's equations in gravitational field
Field tensor:
\[\begin{align}
F_{ik}=A_{k;i}-A_{i;k}=\frac{\partial A_k}{\partial x^i}-\frac{\partial A_i}{\partial x^k}
\end{align}\]
Four electric current density:
Preliminaries:
Invariant four-volume:
\[\begin{align}
d\Omega=dx'^0dx'^1dx'^2dx'^3=\frac{\partial(x'^0,x'^1,x'^2,x'^3)}{\partial(x^0,x^1,x^2,x^3)}dx^0dx^1dx^2dx^3
\end{align}\]
Here
\(J=\frac{\partial(x'^0,x'^1,x'^2,x'^3)}{\partial(x^0,x^1,x^2,x^3)}\)
stands for the
Jacobian, i.e., determinant
of the transformation
matrix \(\partial
x'^i/\partial x^j\).
Consider
\[\begin{align}
g'_{ij}=\frac{\partial
x^k}{\partial
x'^i}\frac{\partial
x^l}{\partial
x'^j}g_{kl}
\end{align}\]
Taking the determinant
of both sides, we have
\[\begin{align}
-1=\frac{1}{J^2}g\;,
\end{align}\]
hence
\[\begin{align}
J=\sqrt{-g}\;,
\end{align}\]
and
\[\begin{align}
d\Omega=\sqrt{-g}dx^0dx^1dx^2dx^3
\end{align}\]
Similarly, the proper three-volume element is
\[\begin{align}
\sqrt{\gamma}dx^1dx^2dx^3\;,
\end{align}\]
where \(\gamma\) stands for the determinant of the three-metric \(-g_{\alpha\beta}+g_{0\alpha}g_{0\beta}/g_{00}\).
Relation between the determinants \(g\) and \(\gamma\):
\[\begin{align}
g=\text{det}
\begin{pmatrix}
g_{00} && g_{0\beta}\\
g_{\alpha 0} && g_{\alpha\beta}
\end{pmatrix}
=\text{det}
\begin{pmatrix}
g_{00} && 0\\
g_{\alpha 0} && g_{\alpha\beta}-g_{\alpha 0}g_{0\beta}/g_{00}
\end{pmatrix}
=g_{00}\left(-\gamma\right)
\end{align}\]
(end of preliminaries)
\[\begin{align}
j^i=\frac{de\;c\;dx^i}{d\Omega}=\frac{de\;c\;dx^i}{\sqrt{-g}dx^0dx^1dx^2dx^3}=\frac{de\;c}{\sqrt{\gamma}dx^1dx^2dx^3}\frac{dx^i}{\sqrt{g_{00}}dx^0}=\frac{\rho c}{\sqrt{g_{00}}}\frac{dx^i}{dx^0}
\end{align}\]
Maxwell's equations:
\[\begin{align}
\frac{\partial F_{ik}}{\partial x^l}+\frac{\partial F_{li}}{\partial x^k}+\frac{\partial F_{kl}}{\partial x^i}=0
\end{align}\]
\[\begin{align}
F^{ik}_{;k}=\frac{1}{\sqrt{-g}}\frac{\partial }{\partial x^k}\left(\sqrt{-g}F^{ik}\right)=-\frac{j^i}{\epsilon_0 c^2}
\end{align}\]
Continuity equation for electric charge:
\[\begin{align}
j^i_{;i}=\frac{1}{\sqrt{-g}}\frac{\partial }{\partial x^i}\left(\sqrt{-g}\;j^{i}\right)=0
\end{align}\]
Proof of divergence
expressions:
Note that
\[\begin{align}
\frac{\partial g}{\partial x^i}=\frac{\partial g_{jk}}{\partial x^i}\left(g\;g^{jk}\right)
\end{align}\]
Then
\[\begin{align}
F^{ik}_{;k}=F^{ik}_{,k}+\underbrace{\Gamma^i_{\phantom{i}mk}F^{mk}}_{=0}+\Gamma^k_{\phantom{k}mk}F^{im}
\end{align}\]
Here
\[\begin{align}
\Gamma^k_{\phantom{k}mk}=\frac{1}{2}g^{jk}\left(g_{jm,k}+g_{jk,m}-g_{mk,j}\right)=\frac{1}{2}g^{jk}g_{jk,m}=\frac{1}{2g}\frac{\partial g}{\partial x^m}=\frac{1}{\sqrt{-g}}\frac{\partial \sqrt{-g}}{\partial x^m}\;.
\end{align}\]
Therefore
\[\begin{align}
F^{ik}_{;k}=F^{ik}_{,k}+\frac{1}{\sqrt{-g}}\frac{\partial \sqrt{-g}}{\partial x^m}F^{im}=\frac{1}{\sqrt{-g}}\frac{\partial }{\partial x^k}\left(\sqrt{-g}F^{ik}\right)
\end{align}\]
Similarly
\[\begin{align}
j^{i}_{;i}=j^{i}_{,i}+\Gamma^i_{\phantom{i}mi}j^m=j^{i}_{,i}+\frac{1}{\sqrt{-g}}\frac{\partial \sqrt{-g}}{\partial x^m}j^{m}=\frac{1}{\sqrt{-g}}\frac{\partial }{\partial x^i}\left(\sqrt{-g}\;j^{i}\right)
\end{align}\]
(end of proof)
Motion of charged particle in electromagnetic and gravitational fields:
\[\begin{align}
m\left(\frac{d u^i}{ds}+\Gamma^i_{\phantom{1}kl}u^k u^l\right)=qF^{ik}u_k
\end{align}\]
bene@arpad.elte.hu
