# 3: Relativity


## Special relativity

### The Lorentz transformation

The Lorentz transform $$(\vec{x}\,',t')=(\vec{x}\,'(\vec{x},t),t'(\vec{x},t))$$ 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 transform can also be found when $$ds^2=ds'^2$$ is demanded. The general form of the Lorentz transform 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 $$\vec{v}=v\vec{e}_x$$:

$p'_x=\gamma\left(p_x-\frac{\beta W}{c}\right)~,~~~W'=\gamma(W-vp_x)$ 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 reference frame moving parallel at the same velocity 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 transform 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 non-zero rest mass: $$U^\alpha U_\alpha=-c^2$$, for particles with zero rest mass (so with $$v=c$$) then $$U^\alpha U_\alpha=0$$. The 4-vector for energy and momentum is given by: $$p^\alpha=m_0U^\alpha=(\gamma p^i,iW/c)$$. So: $$p_\alpha p^\alpha=-m_0^2c^2=p^2-W^2/c^2$$.

### Red and blue shift

There are three causes of red and blue shifts:

1. Motion: with $$\vec{e}_v\cdot\vec{e}_r=\cos(\varphi)$$ from which 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: $$\displaystyle\frac{\lambda_0}{\lambda_1}=\frac{R_0}{R_1}$$.

### 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 then 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

### Riemann 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 where 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 the variation $$g_{\mu\nu}\rightarrow g_{\mu\nu}+\delta g_{\mu\nu}$$ from which the Einstein field equations can be derived:

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

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 for 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.

### 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.

### 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 also be found in the classical case 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 a minimum or $$u$$ a maximum. 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.

### The trajectory of a photon

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

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

### 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 if the relative velocities are $$\ll c$$ and the wavelengths $$\gg$$ than the size of the system, the loss of energy of a mechanical 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.

### Cosmology

If for the universe as a whole it 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 scale factor $$R(t)$$ 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 three 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 associated 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.
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This page titled 3: Relativity is shared under a CC BY license and was authored, remixed, and/or curated by Johan Wevers.