The Lorentz transformation (→x′,t′)=(→x′(→x,t),t′(→x,t)) leaves the wave equation invariant if c is invariant:
∂2∂x2+∂2∂y2+∂2∂z2−1c2∂2∂t2=∂2∂x′2+∂2∂y′2+∂2∂z′2−1c2∂2∂t′2
This transformation can also be found when ds2=ds′2 is demanded. The general form of the Lorentz transformation is given by:
→x′=→x+(γ−1)(→x⋅→v)→v|v|2−γ→vt , t′=γ(t−→x⋅→vc2)
where
γ=1√1−v2c2
The velocity difference →v′ between two observers transforms according to:
→v′=(γ(1−→v1⋅→v2c2))−1(→v2+(γ−1)→v1⋅→v2v21→v1−γ→v1)
If the velocity is parallel to the x-axis, this becomes y′=y, z′=z and:
x′=γ(x−vt) , x=γ(x′+vt′)t′=γ(t−xvc2) , t=γ(t′+x′vc2) , v′=v2−v11−v1v2c2
If →v=v→ex holds:
p′x=γ(px−βWc) , W′=γ(W−vpx) With β=v/c the electric field of a moving charge is given by:
→E=Q4πε0r2(1−β2)→er(1−β2sin2(θ))3/2
The electromagnetic field transforms according to:
→E′=γ(→E+→v×→B) , →B′=γ(→B−→v×→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=d→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 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=p2−W2/c2.
There are three causes of red and blue shifts:
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=∂Aβ∂xα−∂Aα∂xβ
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
The basic principles of general relativity are:
The Riemann tensor is defined as: RμναβTν:=∇α∇βTμ−∇β∇αTμ, where the covariant derivative is given by ∇jai=∂jai+Γijkak and ∇jai=∂jai−Γkijak. Here,
Γijk=gil2(∂glj∂xk+∂glk∂xj−∂gjk∂xl)
for Euclidean spaces this reduces to:
Γijk=∂2ˉxl∂xj∂xk∂xi∂ˉxl,
where Γijk are the Christoffel symbols. For a second-order tensor [∇α,∇β]Tμν=RμσαβTσν+RσναβTμσ, ∇kaij=∂kaij−Γlkjail+Γiklalj, ∇kaij=∂kaij−Γlkialj−Γlkjajl and ∇kaij=∂kaij+Γiklalj+Γjklail. The following holds: Rαβμν=∂μΓαβν−∂νΓαβμ+ΓασμΓσβν−ΓασνΓσβμ.
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:=Rαβ−12gαβR, 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 gμν→gμν+δgμν the Einstein field equations can be derived:
Gαβ=8πκc2Tαβ, which can also be written asRαβ=8πκc2(Tαβ−12gαβTμμ)
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: Rαβ−12gαβR+Λgαβ=8πκc2Tαβ
where Λ is the cosmological constant. This constant plays a role in inflatory models of the universe.
The metric tensor in an Euclidean space is given by: gij=∑k∂ˉxk∂xi∂ˉxk∂xj.
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+(1−2mr)−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+(1−2V)(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=−32m3re−r/2m(dv2−du2)+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 v2−u2=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=(1−2mr−e2r2+a2cos2θ)c2dt2−(r2+a2cos2θr2−2mr+a2−e2)dr2−(r2+a2cos2θ)dθ2−(r2+a2+(2mr−e2)a2sin2θr2+a2cos2θ)sin2θdφ2+(2a(2mr−e2)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+√m2−a2−e2.
Near rotating black holes frame dragging occurs because gtφ≠0. For the Kerr metric (e=0, a≠0) then follows that within the surface RE=m+√m2−a2cos2θ (the ergosphere) no particle can be at rest.
To find a planetary orbit, the variational problem δ∫ds=0 has to be solved. This is equivalent to the problem δ∫ds2=δ∫gijdxidxj=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(φ−εφ)+ε(A+B22A−B26Acos(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(φ−εφ)=0⇒φ≈2πn(1+ε). For the perihelion shift it then follows that: Δφ=2πε=6πm2/h2 per orbit.
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+u−3mu)=0
Starting with the approximation gμν=ημν+hμν for weak gravitational fields and the definition h′μν=hμν−12ημνhαα it follows that ◻h′μν=0 if the gauge condition ∂h′μν/∂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=−G5c5∑i,j(d3Qijdt3)2
with Qij=∫ϱ(xixj−13δijr2)d3x the mass quadrupole moment.
If for the universe as a whole is assumed:
then the Robertson-Walker metric can be derived for the line element:
ds2=−c2dt2+R2(t)r20(1−kr24r20)(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) km⋅s−1⋅Mpc−1. This gives 3 possible conditions for the universe (here, W is the total amount of energy in the universe):