3. Relativity

3.1 Special relativity

3.1.1 The Lorentz transformation

The Lorentz transformation $(\vec{x}\,',t')=(\vec{x}\,'(\vec{x},t),t'(\vec{x},t))$ leaves the wave equation invariant if $c$ is invariant:

$$\frac{\partial^2 }{\partial x^2}+\frac{\partial^2 }{\partial y^2}+\frac{\partial^2 }{\partial z^2}-\frac{1}{c^2}\frac{\partial^2 }{\partial t^2}= \frac{\partial^2 }{\partial x'^2}+\frac{\partial^2 }{\partial y'^2}+\frac{\partial^2 }{\partial z'^2}-\frac{1}{c^2}\frac{\partial^2 }{\partial t'^2}$$

This transformation can also be found when $ds^2=ds'^2$ is demanded. The general form of the Lorentz transformation is given by: 

$$\vec{x}\;'=\vec{x} + \frac{(\gamma-1)(\vec{x}\cdot \vec{v}\; ) \vec{v}}{|v|^2}-\gamma\vec{v}t  ~~,~~ t'=\gamma\left(t-\frac{\vec{x}\cdot\vec{v}}{c^2}\right)$$

where

$$\gamma=\frac{1}{\sqrt{1-\frac{\displaystyle v^2}{\displaystyle c^2}}}$$

The velocity difference $\vec{v}\,'$ between two observers transforms according to:

$$\vec{v}\,'=\left(\gamma\left(1-\frac{\vec{v}_1\cdot\vec{v}_2}{c^2}\right)\right)^{-1} \left(\vec{v}_2+(\gamma-1)\frac{\vec{v}_1\cdot\vec{v}_2}{v_1^2}\vec{v}_1-\gamma\vec{v}_1\right)$$

If the velocity is parallel to the $x$-axis, this becomes $y'=y$, $z'=z$ and:

$$\begin{aligned} &&x'=\gamma(x-vt)~,~~~x=\gamma(x'+vt')\\ &&t'=\gamma\left(t-\displaystyle\frac{xv}{c^2}\right)~,~~~t=\gamma\left(t'+\displaystyle\frac{x'v}{c^2}\right)~,~~~ v'=\frac{\displaystyle v_2-v_1}{\displaystyle 1-\frac{v_1v_2}{c^2}}\end{aligned}$$

If $\vec{v}=v\vec{e}_x$ holds:

$$p'_x=\gamma\left(p_x-\frac{\beta W}{c}\right)~,~~~W'=\gamma(W-vp_x)$$ With $\beta=v/c$ the electric field of a moving charge is given by:

$$\vec{E}=\frac{Q}{4\pi\varepsilon_0r^2}\frac{(1-\beta^2)\vec{e}_{r}}{(1-\beta^2\sin^2(\theta))^{3/2}}$$

The electromagnetic field transforms according to:

$$\vec{E}'=\gamma(\vec{E}+\vec{v}\times\vec{B}\,)~~,~~~ \vec{B}'=\gamma\left(\vec{B}-\frac{\vec{v}\times\vec{E}}{c^2}\right)$$

Length, mass and time transform according to: $\Delta t_{\rm r}=\gamma\Delta t_{\rm 0}$, $m_{\rm r}=\gamma m_0$, $l_{\rm r}=l_0/\gamma$, with ${\rm 0}$ labeling the quantities in a co-moving reference frame and ${\rm r}$ labeling the quantities in a frame moving with velocity $v$ w.r.t. it. The proper time $\tau$ is defined as: $d\tau^2=ds^2/c^2$, so $\Delta\tau=\Delta t/\gamma$. Energy and momentum are: $W=m_{\rm r}c^2=\gamma W_0$, $W^2=m_0^2c^4+p^2c^2$. $p=m_{\rm r}v=\gamma m_0v=Wv/c^2$, and $pc=W\beta$ where $\beta=v/c$. The force is defined by $\vec{F}=d\vec{p}/dt$.

4-vectors have the property that their modulus is independent of the observer: their components can change after a coordinate transformation but not their modulus. The difference of two 4-vectors transforms also as a 4-vector. The 4-vector for the velocity is given by $\displaystyle U^\alpha=\frac{dx^\alpha}{d\tau}$. The relation with the “common” velocity $u^i:=dx^i/dt$ is: $U^\alpha=(\gamma u^i,ic\gamma)$. For particles with nonzero restmass: $U^\alpha U_\alpha=-c^2$, for particles with zero restmass (so with $v=c$) : $U^\alpha U_\alpha=0$. The 4-vector for energy and momentum is given by: $p^\alpha=m_0U^\alpha=(\gamma p^i,iW/c)$. So: $p_\alpha p^\alpha=-m_0^2c^2=p^2-W^2/c^2$.

3.1.2 Red and blue shift

There are three causes of red and blue shifts:

1. Motion: with $\vec{e}_v\cdot\vec{e}_r=\cos(\varphi)$ follows: $\displaystyle \frac{f'}{f}=\gamma\left(1-\frac{v\cos(\varphi)}{c}\right)$.
   This can give both red- and blueshift, also $\perp$ to the direction of motion.
2. Gravitational redshift: $\displaystyle\frac{\Delta f}{f}=\frac{\kappa M}{rc^2}$.
3. Redshift because the universe expands, resulting in e.g. the cosmic background radiation: $\displaystyle\frac{\lambda_0}{\lambda_1}=\frac{R_0}{R_1}$.

3.1.3 The stress-energy tensor and the field tensor

The stress-energy tensor is given by:

$$T_{\mu\nu}=(\varrho c^2+p)u_\mu u_\nu+pg_{\mu\nu}+\frac{1}{c^2} \left(F_{\mu\alpha}F^\alpha_\nu+\mbox{$\frac{1}{4}$}g_{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}\right)$$

The conservation laws can than be written as: $\nabla_\nu T^{\mu\nu}=0$. The electromagnetic field tensor is given by:

$$F_{\alpha\beta}=\frac{\partial A_\beta}{\partial x^\alpha}-\frac{\partial A_\alpha}{\partial x^\beta}$$

with $A_\mu:=(\vec{A},iV/c)$ and $J_\mu:=(\vec{J},ic\rho)$. Maxwell's equations can than be written as:

$$\partial_\nu F^{\mu\nu}=\mu_0J^\mu~,~~ \partial_\lambda F_{\mu\nu}+\partial_\mu F_{\nu\lambda}+\partial_\nu F_{\lambda\mu}=0$$

The equations of motion for a charged particle in an EM field become with the field tensor:

$$\frac{dp_\alpha}{d\tau}=qF_{\alpha\beta}u^\beta$$

General relativity

3.2.1 Riemannian geometry, the Einstein tensor

The basic principles of general relativity are:

1. The geodesic postulate: free falling particles move along geodesics of space-time with the proper time $\tau$ or arc length $s$ as parameter. For particles with zero rest mass (photons), the use of a free parameter is required because for them $ds=0$. From $\delta\int ds=0$ the equations of motion can be derived:
   $$\frac{d^2x^\alpha}{ds^2}+\Gamma_{\beta\gamma}^{\alpha}\frac{dx^\beta}{ds}\frac{dx^\gamma}{ds}=0$$
2. The principle of equivalence: inertial mass $\equiv$ gravitational mass $\Rightarrow$ gravitation is equivalent with a curved space-time were particles move along geodesics.
3. By a proper choice of the coordinate system it is possible to make the metric locally flat in each point $x_i$: $g_{\alpha\beta}(x_i)=\eta_{\alpha\beta}:=$diag$(-1,1,1,1)$.

The Riemann tensor is defined as: $R^\mu_{\nu\alpha\beta}T^\nu:=\nabla_\alpha\nabla_\beta T^\mu-\nabla_\beta\nabla_\alpha T^\mu$, where the covariant derivative is given by $\nabla_j a^i=\partial_ja^i+\Gamma_{jk}^ia^k$ and $\nabla_j a_i=\partial_ja_i-\Gamma_{ij}^ka_k$. Here,

$$\Gamma_{jk}^i=\frac{g^{il}}{2}\left(\frac{\partial g_{lj}}{\partial x^k}+ \frac{\partial g_{lk}}{\partial x^j}-\frac{\partial g_{_jk}}{\partial x^l}\right)$$  

for Euclidean spaces this reduces to:

$$\Gamma_{jk}^i=\frac{\partial^2\bar{x}^l}{\partial x^j\partial x^k}\frac{\partial x^i}{\partial \bar{x}^l},$$

where $\Gamma_{jk}^i$ are the Christoffel symbols. For a second-order tensor $[\nabla_\alpha,\nabla_\beta]T_\nu^\mu=R_{\sigma\alpha\beta}^\mu T^\sigma_\nu+R^\sigma_{\nu\alpha\beta}T^\mu_\sigma$, $\nabla_k a^i_j=\partial_ka^i_j-\Gamma_{kj}^la_l^i+\Gamma_{kl}^ia_j^l$, $\nabla_k a_{ij}=\partial_ka_{ij}-\Gamma_{ki}^la_{lj}-\Gamma_{kj}^la_{jl}$ and $\nabla_k a^{ij}=\partial_ka^{ij}+\Gamma_{kl}^ia^{lj}+\Gamma_{kl}^ja^{il}$. The following holds: $R_{\beta\mu\nu}^\alpha=\partial_\mu\Gamma_{\beta\nu}^\alpha-\partial_\nu\Gamma_{\beta\mu}^\alpha+ \Gamma_{\sigma\mu}^\alpha\Gamma_{\beta\nu}^\sigma-\Gamma_{\sigma\nu}^\alpha\Gamma_{\beta\mu}^\sigma$.

The Ricci tensor is a contraction of the Riemann tensor: $R_{\alpha\beta}:=R^\mu_{\alpha\mu\beta}$, which is symmetric: $R_{\alpha\beta}=R_{\beta\alpha}$. The Bianchi identities are: $\nabla_\lambda R_{\alpha\beta\mu\nu}+\nabla_\nu R_{\alpha\beta\lambda\mu}+ \nabla_\mu R_{\alpha\beta\nu\lambda}=0$.

The Einstein tensor is given by: $G^{\alpha\beta}:=R^{\alpha\beta}-\frac{1}{2}g^{\alpha\beta}R$, where $R:=R_\alpha^\alpha$ is the Ricci scalar, for which: $\nabla_\beta G_{\alpha\beta}=0$. With the variational principle $\delta\int({\cal L}(g_{\mu\nu})-Rc^2/16\pi\kappa)\sqrt{|g|}d^4x=0$ for variations $g_{\mu\nu}\rightarrow g_{\mu\nu}+\delta g_{\mu\nu}$ the Einstein field equations can be derived:

$$G_{\alpha\beta}=\frac{8\pi\kappa}{c^2}T_{\alpha\beta} \;\;\;\textrm{, which can also be written as}\;\;\;R_{\alpha\beta}=\frac{8\pi\kappa}{c^2}(T_{\alpha\beta}-\mbox{$\frac{1}{2}$}g_{\alpha\beta}T^{\mu}_{\mu})$$

For empty space this is equivalent to $R_{\alpha\beta}=0$. The equation $R_{\alpha\beta\mu\nu}=0$ has as its only solution a flat space.

The Einstein equations are 10 independent equations, which are of second order in $g_{\mu\nu}$. From this, the Laplace equation from Newtonian gravitation can be derived by starting with: $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, where $|h|\ll1$. In the stationary case, this results in $\nabla^2 h_{00}=8\pi\kappa\varrho/c^2$.

The most general form of the field equations is: $\displaystyle R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R+\Lambda g_{\alpha\beta}=\frac{8\pi\kappa}{c^2}T_{\alpha\beta}$

where $\Lambda$ is the cosmological constant. This constant plays a role in inflatory models of the universe.

3.2.2 The line element

The metric tensor in an Euclidean space is given by: $\displaystyle g_{ij}=\sum_k\frac{\partial \bar{x}^k}{\partial x^i}\frac{\partial \bar{x}^k}{\partial x^j}$.

In general: $ds^2=g_{\mu\nu}dx^\mu dx^\nu$ holds. In special relativity this becomes $ds^2=-c^2dt^2+dx^2+dy^2+dz^2$. This metric, $\eta_{\mu\nu}:=$diag$(-1,1,1,1)$, is called the Minkowski metric.

The external Schwarzschild metric applies in vacuum outside a spherical mass distribution, and is given by:

$$ds^2=\left(-1+\frac{2m}{r}\right)c^2dt^2+\left(1-\frac{2m}{r}\right)^{-1}dr^2+r^2d\Omega^2$$

Here, $m:=M\kappa/c^2$ is the geometrical mass of an object with mass $M$, and $d\Omega^2=d\theta^2+\sin^2\theta d\varphi^2$. This metric is singular for $r=2m=2\kappa M/c^2$. If an object is smaller than its event horizon $2m$, that implies that its escape velocity is $>c$, it is called a black hole. The Newtonian limit of this metric is given by:

$$ds^2=-(1+2V)c^2dt^2+(1-2V)(dx^2+dy^2+dz^2)$$

where $V=-\kappa M/r$ is the Newtonian gravitation potential. In general relativity, the components of $g_{\mu\nu}$ are associated with the potentials and the derivatives of $g_{\mu\nu}$ with the field strength.

The Kruskal-Szekeres coordinates are used to solve certain problems with the Schwarzschild metric near $r=2m$. They are defined by:

• $r>2m$: $$\left\{\begin{array}{ccl} u&=&\displaystyle{\sqrt{\frac{r}{2m}-1}\exp\left(\frac{r}{4m}\right)\cosh\left(\frac{t}{4m}\right)}\\ &&\\ v&=&\displaystyle{\sqrt{\frac{r}{2m}-1}\exp\left(\frac{r}{4m}\right)\sinh\left(\frac{t}{4m}\right)} \end{array}\right.$$
• $r<2m$: $$\left\{\begin{array}{ccl} u&=&\displaystyle{\sqrt{1-\frac{r}{2m}}\exp\left(\frac{r}{4m}\right)\sinh\left(\frac{t}{4m}\right)}\\ &&\\ v&=&\displaystyle{\sqrt{1-\frac{r}{2m}}\exp\left(\frac{r}{4m}\right)\cosh\left(\frac{t}{4m}\right)} \end{array}\right.$$
• $r=2m$: here, the Kruskal coordinates are singular, which is necessary to eliminate the coordinate singularity there.

The line element in these coordinates is given by:

$$ds^2=-\frac{32m^3}{r}{\rm e}^{-r/2m}(dv^2-du^2)+r^2d\Omega^2$$

The line $r=2m$ corresponds to $u=v=0$, the limit $x^0\rightarrow\infty$ with $u=v$ and $x^0\rightarrow-\infty$ with $u=-v$. The Kruskal coordinates are only singular on the hyperbola $v^2-u^2=1$, this corresponds with $r=0$. On the line $dv=\pm du$  $d\theta=d\varphi=ds=0$ holds.

For the metric outside a rotating, charged spherical mass the Newman metric applies:

$$\begin{aligned} ds^2&=&\left(1-\frac{2mr-e^2}{r^2+a^2\cos^2\theta}\right)c^2dt^2- \left(\frac{r^2+a^2\cos^2\theta}{r^2-2mr+a^2-e^2}\right)dr^2- (r^2+a^2\cos^2\theta)d\theta^2-\\ &&\left(r^2+a^2+\frac{(2mr-e^2)a^2\sin^2\theta}{r^2+a^2\cos^2\theta}\right)\sin^2\theta d\varphi^2+ \left(\frac{2a(2mr-e^2)}{r^2+a^2\cos^2\theta}\right)\sin^2\theta(d\varphi)(cdt)\end{aligned}$$

where $m=\kappa M/c^2$, $a=L/Mc$ and $e=\kappa Q/\varepsilon_0c^2$.

A rotating charged black hole has an event horizon with $R_{\rm S}=m+\sqrt{m^2-a^2-e^2}$.

Near rotating black holes frame dragging occurs because $g_{t\varphi}\neq0$. For the Kerr metric ($e=0$, $a\neq0$) then follows that within the surface $R_{\rm E}=m+\sqrt{m^2-a^2\cos^2\theta}$ (the ergosphere) no particle can be at rest.

3.2.3 Planetary orbits and the perihelion shift

To find a planetary orbit, the variational problem $\delta\int ds=0$ has to be solved. This is equivalent to the problem $\delta\int ds^2=\delta\int g_{ij}dx^idx^j=0$. Substituting the external Schwarzschild metric yields for a planetary orbit:

$$\frac{du}{d\varphi}\left(\frac{d^2u}{d\varphi^2}+u\right)=\frac{du}{d\varphi}\left(3mu+\frac{m}{h^2}\right)$$

where $u:=1/r$ and $h=r^2\dot{\varphi}=$constant. The term $3mu$ is not present in the classical solution. This term can in the classical case also be found from a potential $\displaystyle V(r)=-\frac{\kappa M}{r}\left(1+\frac{h^2}{r^2}\right)$.

The orbital equation gives $r=$constant as solution, or can, after dividing by $du/d\varphi$, be solved with perturbation theory. In zeroth order, this results in an elliptical orbit: $u_0(\varphi)=A+B\cos(\varphi)$ with $A=m/h^2$ and $B$ an arbitrary constant. In first order, this becomes:

$$u_1(\varphi)=A+B\cos(\varphi-\varepsilon\varphi)+\varepsilon \left(A+\frac{B^2}{2A}-\frac{B^2}{6A}\cos(2\varphi)\right)$$

where $\varepsilon=3m^2/h^2$ is small. The perihelion of a planet is the point for which $r$ is minimal, or $u$ maximal. This is the case if $\cos(\varphi-\varepsilon\varphi)=0\Rightarrow\varphi\approx2\pi n(1+\varepsilon)$. For the perihelion shift it then follows that: $\Delta\varphi=2\pi\varepsilon=6\pi m^2/h^2$ per orbit.

3.2.4 The trajectory of a photon

For the trajectory of a photon (and for each particle with zero restmass) $ds^2=0$. Substituting the external Schwarzschild metric results in the following orbital equation:

$$\frac{du}{d\varphi}\left(\frac{d^2u}{d\varphi^2}+u-3mu\right)=0$$

3.2.5 Gravitational waves

Starting with the approximation $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ for weak gravitational fields and the definition $h'_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h^{\alpha}_{\alpha}$ it follows that $\Box h'_{\mu\nu}=0$ if the gauge condition $\partial h'_{\mu\nu}/\partial x^\nu=0$ is satisfied. From this, it follows that the loss of energy of a mechanical system, if the occurring velocities are $\ll c$ and for wavelengths $\gg$ the size of the system, is given by:

$$\frac{dE}{dt}=-\frac{G}{5c^5}\sum_{i,j}\left(\frac{d^3Q_{ij}}{dt^3}\right)^2$$

with $Q_{ij}=\int\varrho(x_ix_j-\frac{1}{3}\delta_{ij}r^2)d^3x$ the mass quadrupole moment.

3.2.6 Cosmology

If for the universe as a whole is assumed:

1. There exists a global time coordinate which acts as $x^0$ of a Gaussian coordinate system,
2. The 3-dimensional spaces are isotrope for a certain value of $x^0$,
3. Each point is equivalent to each other point for a fixed $x^0$.

then the Robertson-Walker metric can be derived for the line element:

$$ds^2=-c^2dt^2+\frac{R^2(t)}{r_0^2\left(1-\displaystyle\frac{kr^2}{4r_0^2}\right)}(dr^2+r^2d\Omega^2)$$

For the scalefactor $R(t)$ the following equations can be derived:

$$\frac{2\ddot{R}}{R}+\frac{\dot{R}^2+kc^2}{R^2}=-\frac{8\pi\kappa p}{c^2}+\Lambda ~~~\mbox{and}~~~ \frac{\dot{R}^2+kc^2}{R^2}=\frac{8\pi\kappa\varrho}{3}+\frac{\Lambda}{3}$$

where $p$ is the pressure and $\varrho$ the density of the universe. If $\Lambda=0$ the deceleration parameter $q$ can be derived:

$$q=-\frac{\ddot{R}R}{\dot{R}^2}=\frac{4\pi\kappa\varrho}{3H^2}$$

where $H=\dot{R}/R$ is Hubble’s constant. This is a measure of the velocity with which galaxies far away are moving away from each other, and has the value $\approx(75\pm25)$ km$\cdot$s$^{-1}\cdot$Mpc$^{-1}$. This gives 3 possible conditions for the universe (here, $W$ is the total amount of energy in the universe):

1. Parabolical universe: $k=0$, $W=0$, $q=\frac{1}{2}$. The expansion velocity of the universe $\rightarrow0$ if $t\rightarrow\infty$. The hereto related critical density is $\varrho_{\rm c}=3H^2/8\pi\kappa$.
2. Hyperbolical universe: $k=-1$, $W<0$, $q<\frac{1}{2}$. The expansion velocity of the universe remains positive forever.
3. Elliptical universe: $k=1$, $W>0$, $q>\frac{1}{2}$. The expansion velocity of the universe becomes negative after some time: the universe starts collapsing.