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3. Relativity

3.1 Special relativity

3.1.1 The Lorentz transformation

The Lorentz transformation (x,t)=(x(x,t),t(x,t)) leaves the wave equation invariant if c is invariant:

2x2+2y2+2z21c22t2=2x2+2y2+2z21c22t2

This transformation can also be found when ds2=ds2 is demanded. The general form of the Lorentz transformation is given by: 

x=x+(γ1)(xv)v|v|2γvt  ,  t=γ(txvc2)

where

γ=11v2c2

The velocity difference v between two observers transforms according to:

v=(γ(1v1v2c2))1(v2+(γ1)v1v2v21v1γv1)

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

x=γ(xvt) ,   x=γ(x+vt)t=γ(txvc2) ,   t=γ(t+xvc2) ,   v=v2v11v1v2c2

If v=vex holds:

px=γ(pxβWc) ,   W=γ(Wvpx) With β=v/c the electric field of a moving charge is given by:

E=Q4πε0r2(1β2)er(1β2sin2(θ))3/2

The electromagnetic field transforms according to:

E=γ(E+v×B)  ,   B=γ(Bv×Ec2)

Length, mass and time transform according to: Δtr=γΔt0, mr=γm0, lr=l0/γ, with 0 labeling the quantities in a co-moving reference frame and r labeling the quantities in a frame moving with velocity v w.r.t. it. The proper time τ is defined as: dτ2=ds2/c2, so Δτ=Δt/γ. Energy and momentum are: W=mrc2=γW0, W2=m20c4+p2c2. p=mrv=γm0v=Wv/c2, and pc=Wβ where β=v/c. The force is defined by F=dp/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 Uα=dxαdτ. The relation with the “common” velocity ui:=dxi/dt is: Uα=(γui,icγ). For particles with nonzero restmass: UαUα=c2, for particles with zero restmass (so with v=c) : UαUα=0. The 4-vector for energy and momentum is given by: pα=m0Uα=(γpi,iW/c). So: pαpα=m20c2=p2W2/c2.

3.1.2 Red and blue shift

There are three causes of red and blue shifts:

  1. Motion: with ever=cos(φ) follows: ff=γ(1vcos(φ)c).
    This can give both red- and blueshift, also to the direction of motion.
  2. Gravitational redshift: Δff=κMrc2.
  3. Redshift because the universe expands, resulting in e.g. the cosmic background radiation: λ0λ1=R0R1.

3.1.3 The stress-energy tensor and the field tensor

The stress-energy tensor is given by:

Tμν=(ϱc2+p)uμuν+pgμν+1c2(FμαFαν+14gμνFαβFαβ)

The conservation laws can than be written as: νTμν=0. The electromagnetic field tensor is given by:

Fαβ=AβxαAαxβ

with Aμ:=(A,iV/c) and Jμ:=(J,icρ). Maxwell's equations can than be written as:

νFμν=μ0Jμ ,  λFμν+μFνλ+νFλμ=0

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

dpαdτ=qFαβuβ

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 τ 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 δds=0 the equations of motion can be derived:
    d2xαds2+Γαβγdxβdsdxγds=0
  2. The principle of equivalence: inertial mass gravitational mass 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 xi: gαβ(xi)=ηαβ:=diag(1,1,1,1).

The Riemann tensor is defined as: RμναβTν:=αβTμβαTμ, where the covariant derivative is given by jai=jai+Γijkak and jai=jaiΓkijak. Here,

Γijk=gil2(gljxk+glkxjgjkxl) 

for Euclidean spaces this reduces to:

Γijk=2ˉxlxjxkxiˉxl,

where Γijk are the Christoffel symbols. For a second-order tensor [α,β]Tμν=RμσαβTσν+RσναβTμσ, kaij=kaijΓlkjail+Γiklalj, kaij=kaijΓlkialjΓlkjajl and kaij=kaij+Γiklalj+Γjklail. The following holds: Rαβμν=μΓαβννΓαβμ+ΓασμΓσβνΓασνΓσβμ.

The Ricci tensor is a contraction of the Riemann tensor: Rαβ:=Rμαμβ, which is symmetric: Rαβ=Rβα. The Bianchi identities are: λRαβμν+νRαβλμ+μRαβνλ=0.

The Einstein tensor is given by: Gαβ:=Rαβ12gαβR, where R:=Rαα is the Ricci scalar, for which: βGαβ=0. With the variational principle δ(L(gμν)Rc2/16πκ)|g|d4x=0 for variations gμνgμν+δgμν the Einstein field equations can be derived:

Gαβ=8πκc2Tαβ, which can also be written asRαβ=8πκc2(Tαβ12gαβTμμ)

For empty space this is equivalent to Rαβ=0. The equation Rαβμν=0 has as its only solution a flat space.

The Einstein equations are 10 independent equations, which are of second order in gμν. From this, the Laplace equation from Newtonian gravitation can be derived by starting with: gμν=ημν+hμν, where |h|1. In the stationary case, this results in 2h00=8πκϱ/c2.

The most general form of the field equations is: Rαβ12gαβR+Λgαβ=8πκc2Tαβ

where Λ 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: gij=kˉxkxiˉxkxj.

In general: ds2=gμνdxμdxν holds. In special relativity this becomes ds2=c2dt2+dx2+dy2+dz2. This metric, ημν:=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:

ds2=(1+2mr)c2dt2+(12mr)1dr2+r2dΩ2

Here, m:=Mκ/c2 is the geometrical mass of an object with mass M, and dΩ2=dθ2+sin2θdφ2. This metric is singular for r=2m=2κM/c2. 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:

ds2=(1+2V)c2dt2+(12V)(dx2+dy2+dz2)

where V=κM/r is the Newtonian gravitation potential. In general relativity, the components of gμν are associated with the potentials and the derivatives of gμν 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:

The line element in these coordinates is given by:

ds2=32m3rer/2m(dv2du2)+r2dΩ2

The line r=2m corresponds to u=v=0, the limit x0 with u=v and x0 with u=v. The Kruskal coordinates are only singular on the hyperbola v2u2=1, this corresponds with r=0. On the line dv=±du  dθ=dφ=ds=0 holds.

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

ds2=(12mre2r2+a2cos2θ)c2dt2(r2+a2cos2θr22mr+a2e2)dr2(r2+a2cos2θ)dθ2(r2+a2+(2mre2)a2sin2θr2+a2cos2θ)sin2θdφ2+(2a(2mre2)r2+a2cos2θ)sin2θ(dφ)(cdt)

where m=κM/c2, a=L/Mc and e=κQ/ε0c2.

A rotating charged black hole has an event horizon with RS=m+m2a2e2.

Near rotating black holes frame dragging occurs because gtφ0. For the Kerr metric (e=0, a0) then follows that within the surface RE=m+m2a2cos2θ (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 δds=0 has to be solved. This is equivalent to the problem δds2=δgijdxidxj=0. Substituting the external Schwarzschild metric yields for a planetary orbit:

dudφ(d2udφ2+u)=dudφ(3mu+mh2)

where u:=1/r and h=r2˙φ=constant. The term 3mu is not present in the classical solution. This term can in the classical case also be found from a potential V(r)=κMr(1+h2r2).

The orbital equation gives r=constant as solution, or can, after dividing by du/dφ, be solved with perturbation theory. In zeroth order, this results in an elliptical orbit: u0(φ)=A+Bcos(φ) with A=m/h2 and B an arbitrary constant. In first order, this becomes:

u1(φ)=A+Bcos(φεφ)+ε(A+B22AB26Acos(2φ))

where ε=3m2/h2 is small. The perihelion of a planet is the point for which r is minimal, or u maximal. This is the case if cos(φεφ)=0φ2πn(1+ε). For the perihelion shift it then follows that: Δφ=2πε=6πm2/h2 per orbit.

3.2.4 The trajectory of a photon

For the trajectory of a photon (and for each particle with zero restmass) ds2=0. Substituting the external Schwarzschild metric results in the following orbital equation:

dudφ(d2udφ2+u3mu)=0

3.2.5 Gravitational waves

Starting with the approximation gμν=ημν+hμν for weak gravitational fields and the definition hμν=hμν12ημνhαα it follows that hμν=0 if the gauge condition hμν/xν=0 is satisfied. From this, it follows that the loss of energy of a mechanical system, if the occurring velocities are c and for wavelengths the size of the system, is given by:

dEdt=G5c5i,j(d3Qijdt3)2

with Qij=ϱ(xixj13δijr2)d3x 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 x0 of a Gaussian coordinate system,
  2. The 3-dimensional spaces are isotrope for a certain value of x0,
  3. Each point is equivalent to each other point for a fixed x0.

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

ds2=c2dt2+R2(t)r20(1kr24r20)(dr2+r2dΩ2)

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

2¨RR+˙R2+kc2R2=8πκpc2+Λ   and   ˙R2+kc2R2=8πκϱ3+Λ3

where p is the pressure and ϱ the density of the universe. If Λ=0 the deceleration parameter q can be derived:

q=¨RR˙R2=4πκϱ3H2

where H=˙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 (75±25) kms1Mpc1. 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=12. The expansion velocity of the universe 0 if t. The hereto related critical density is ϱc=3H2/8πκ.
  2. Hyperbolical universe: k=1, W<0, q<12. The expansion velocity of the universe remains positive forever.
  3. Elliptical universe: k=1, W>0, q>12. The expansion velocity of the universe becomes negative after some time: the universe starts collapsing.