© Springer Nature Switzerland AG 2019
M. H. Soffel, W.-B. HanApplied General RelativityAstronomy and Astrophysics Libraryhttps://doi.org/10.1007/978-3-030-19673-8_5

5. Einstein’s Theory of Gravity

Michael H. Soffel1  and Wen-Biao Han2
(1)
Institute of planetary geodesy, Lohrmann-Observatory, Dresden, Germany
(2)
Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai, China
 

5.1 General Relativity

Special Relativity can be described as physics in a 4-dimensional space-time manifold $$\mathcal {M}$$ with metric tensor g μν that reduces to η μν = diag(−1, +1, +1, +1) in any global inertial coordinate system. Such selected global coordinates exist because the geometry of Minkowskian space-time is flat, i.e., the curvature and Ricci tensor vanish. Einstein’s theory of gravity is also a structure $$(\mathcal {M},g)$$ , but space-time geometry in the presence of gravitational fields is not longer flat, the curvature tensor describing the tidal actions. For vanishing gravitational fields the structure $$(\mathcal {M},g)$$ reduces to the Minkowskian space-time; it is fully in accordance with all experiments from Special Relativity.

In General Relativity (GR) all aspects of gravitational fields are contained in the space-time metric tensor. A necessary prerequisite this is the Equivalence principle, that also shows the role of Special Relativity in Einstein’s GT. The weak form of the equivalence principle (the universality of free-fall) has already been discussed. The Einstein equivalence principle (EEP) generalizes this to all non-gravitational laws of physics: in any freely falling system all non-gravitational laws of physics take their form from Special Relativity. In some sense certain aspects of gravity disappear in a freely falling reference frame. Such aspects are related with the affine connections of space-time geometry that are not tensors and can be transformed to zero at any point $$p \in \mathcal {M}$$ by a suitable coordinate transformation. This, however, by no means implies that some existing gravitational field inside such a freely-falling system is zero; if the curvature tensor has non-vanishing components in one coordinate system then there is no coordinate system where it completely vanishes at any point $$p \in \mathcal {M}$$ . This means that the EEP simply says that at each point p of the space-time manifold there are local coordinates such that the metric tensor reduces to the Minkowskian tensor where effects from gravity do not appear.

Einstein’s equivalence principle implies that a reasonable theory of gravity should be a metric theory with $$(\mathcal {M},g)$$ as basic structure and possible additional fields ψ i, taking part in the gravitational interaction. General Relativity is the simplest of all such metric theories, where all additional fields ψ i = 0. Sources of the gravitational field, i.e., all forms of energy and momentum as well as gravity fields itself, produce curvature of space-time which again determines the dynamical behavior of the sources.

5.2 Einstein’s Equivalence Principle

Einstein’s theory of gravity generalizes the results from Minkowski space-time theory by considering also gravitational fields. A hint of how to incorporate gravity into the space-time structure comes from the phenomenon of gravitational redshift. Let us consider two identical clocks at rest in some gravitational potential U(x). Clock 1 is assumed to be located a distance H above clock 1. Then, because of the gravitational redshift the natural frequencies of the two clocks, f 1 and f 2, are related by
$$\displaystyle \begin{aligned} {f_2 \over f_1} = 1 + {1 \over c^2} \left[ U({\mathbf{x}}_2) - U({\mathbf{x}}_1) \right] \, . \end{aligned} $$
(5.2.1)
It is not difficult to see that the gravitational redshift of electromagnetic waves is a consequence of a certain form of the equivalence principle. This will also make it clear why clocks in a gravitational field are running slower (Fig. 5.1).
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Fig. 5.1

Three static clocks in some gravitational field. The larger the gravitational potential the slower the clock runs

Einstein’s Equivalence Principle

Everywhere in the universe and for all times in sufficiently small freely falling laboratories all non-gravitational laws of physics take their form from Special Relativity.

In other words: such freely falling systems are locally inertial. Let us now consider two clocks at rest in some external gravitational field (Fig. 5.2). Obviously the two clocks are not freely falling; instead they are at rest in some system that is accelerated upwards, i.e., away from the center of gravitational attraction. With respect to some freely falling local inertial coordinate system x μ = (ct, x i) the world-lines of the two clocks are depicted in Fig. 5.2. We now consider a light-pulse being emitted from clock 1 in the direction of clock 2. In a first approximation z 1 = gt 2∕2 + H and z 2 = gt 2∕2 and the velocities are given by v = gt. Since in the accelerated system where the two clocks are at rest the situation is stationary and we might choose for simplicity t = 0 for the emission event. Neglecting (v∕c)2-terms t will agree with the proper times indicated by the two clocks. Then for gH∕c 2 ≪ 1 the signal will arrive at clock 2 at t −≃ H∕c. The crucial point is that at the point of reception the second clock has a finite velocity v = gt = gH∕c in the direction of the first clock. Let the first clock emit a second pulse at t = δt 1 immediately after the first one. The arrival time at clock 2 then is
$$\displaystyle \begin{aligned} t_+ = \frac Hc + \delta t_1 - \frac vc \, \delta t_1 \end{aligned}$$
since during the interval δt 1 it has moved a distance v δt 1 in the direction of clock 1, i.e., the effective distance is only H − v δt 1 instead of H. Hence the time that has elapsed during the reception of the two pulses at clock 2 is
$$\displaystyle \begin{aligned} \begin{array}{rcl} \delta t_2 &\displaystyle =&\displaystyle t_+ - t_- = \delta t_1 \, \left( 1 - \frac vc \right) \\ &\displaystyle =&\displaystyle \delta t_1 \, \left( 1 - {g H \over c^2} \right) \simeq \delta t_1 \, \left( 1 - {U({\mathbf{x}}_2) - U({\mathbf{x}}_1) \over c^2} \right) \, \end{array} \end{aligned} $$
in accordance with the gravitational redshift formula (5.2.1). Thus from the standpoint of Einstein’s Equivalence Principle the gravitational redshift results from the first-order Doppler shift of frequencies. Einstein’s form of the equivalence principle has the consequence that gravity can be described by a metric theory, i.e., (see e.g., Will 1993 for more details)

  • by at least a g μν-field and possibly by “other g-fields”;

  • these “other g-fields” only couple to the g μν-field but not to matter-fields directly;

  • at each point in space-time there is a local freely falling system (Einstein’s elevator) where the space-time metric g μν reduces to the flat space-time metric η μν;

  • the world-lines of uncharged test particles are geodesics of g μν.

../images/447007_1_En_5_Chapter/447007_1_En_5_Fig2_HTML.png
Fig. 5.2

Two accelerated clocks as seen from a local freely falling system. The distance between the clocks is H. The first clock emits a first signal, a second one follows after a time interval δt 1. The observer at z 2 receives these two signal at times t − and t +

The last point follows from the fact that these world-lines are straight lines in a freely falling system, i.e., geodesics with respect to the flat space-time metric. Hence, they must be geodesics with respect to the space-time metric g μν.

Metric Property 3

Sufficiently small (uncharged) test bodies move along geodesics of the metric tensor.

The gravitational redshift can then be described in a very elegant manner: we incorporate the gravitational potential U into the metric and write in suitable coordinates
$$\displaystyle \begin{aligned} ds^2 = - \left( 1 - {2U \over c^2} \right)c^2 dt^2 + (d\mathbf{x})^2 \, \end{aligned} $$
(5.2.2)
or
$$\displaystyle \begin{aligned} g_{00} = -1 + {2U \over c^2}; \quad g_{0i} = 0; \quad g_{ij} = \delta_{ij} \, . \end{aligned} $$
(5.2.3)
Assuming again metric property 2 (Eq. (4.​3.​10)) for two clocks at rest (d x = 0) we get for each of the two clocks i:
$$\displaystyle \begin{aligned} d\tau_i^2 = - {1 \over c^2} ds_i^2 = \left( 1 - {2U ({\mathbf{x}}_i) \over c^2} \right) dt^2 \end{aligned}$$
or
$$\displaystyle \begin{aligned} d\tau_i \simeq \left(1 - {U({\mathbf{x}}_i) \over c^2} \right) dt \, . \end{aligned} $$
(5.2.4)
From this we derive
$$\displaystyle \begin{aligned} \begin{array}{rcl} {f_2 \over f_1} &\displaystyle =&\displaystyle {d\tau_1 \over d\tau_2} \simeq {1 - U({\mathbf{x}}_1)/c^2 \over 1 - U({\mathbf{x}}_2)/c^2} \\ &\displaystyle \simeq&\displaystyle 1 + {1 \over c^2} \left[ U({\mathbf{x}}_2) - U({\mathbf{x}}_1) \right] \end{array} \end{aligned} $$
in accordance with (5.2.1).

5.3 The Motion of Test Bodies

Let us consider the geometry that is determined by the metric (5.2.3) in more detail where we restrict our discussion to terms of order c −2. The inverse metric tensor in this approximation is given by
$$\displaystyle \begin{aligned} g^{00} = -1 - {2U \over c^2}; \quad g^{0i} = 0; \quad g^{ij} = \delta_{ij} \, . \end{aligned} $$
(5.3.1)
From this we derive the non-vanishing Christoffel-symbols:
$$\displaystyle \begin{aligned} \Gamma^{0}_{ {0}{i}}{} = \Gamma^{0}_{ {i}{0}}{} = \Gamma^{i}_{ {0}{0}}{} = - {1 \over c^2} U_{,i} \, . \end{aligned} $$
(5.3.2)
We now come to the geodesic equation
$$\displaystyle \begin{aligned} {d^2 x^\mu \over d\lambda^2} + \Gamma^{\mu}_{ {\nu}{\sigma}}{} {dx^\nu \over d\lambda} {dx^\sigma \over d\lambda} = 0 \, . \end{aligned}$$
Here λ is an affine parameter that might be replaced by the time coordinate t (which is not an affine parameter) in the μ = i equation:
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} {d^2 x^i \over dt^2} &\displaystyle =&\displaystyle \left( {dt \over d\lambda} \right)^{-1} {d \over d\lambda} \left[ \left( {dt \over d\lambda} \right)^{-1} {dx^i \over d\lambda} \right] \\ &\displaystyle =&\displaystyle \left( {dt \over d\lambda} \right)^{-2} {d^2 x^i \over d\lambda^2} - \left( {dt \over d\lambda} \right)^{-3} {d^2 t \over d\lambda^2} {dx^i \over d\lambda} \end{array} \end{aligned} $$
(5.3.3)
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle =&\displaystyle - \Gamma^{i}_{ {\nu}{\sigma}}{} {dx^\nu \over dt} {dx^\sigma \over dt} + \frac 1c \Gamma^{0}_{ {\nu}{\sigma}}{} {dx^\nu \over dt} {dx^\sigma \over dt}{dx^i \over dt} \, . \end{array} \end{aligned} $$
(5.3.4)
In more detail this reads (v i ≡ dx i∕dt)
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} {d^2 x^i \over dt^2} = &\displaystyle &\displaystyle - c^2 \left\{ \Gamma^{i}_{ {0}{0}}{} + 2 \Gamma^{i}_{ {0}{j}}{} {v^j \over c} + \Gamma^{i}_{ {j}{k}}{} {v^j \over c} {v^k \over c} \right. \\ &\displaystyle &\displaystyle - \left. \left[ \Gamma^{0}_{ {0}{0}}{} + 2 \Gamma^{0}_{ {0}{j}}{} {v^j \over c} + \Gamma^{0}_{ {j}{k}}{} {v^j \over c} {v^k \over c} \right] {v^i \over c}\right\} \, . \end{array} \end{aligned} $$
(5.3.5)
Considering the Christoffel-symbols from (5.3.2) we see that the right hand side of this equation has a term of order c 0 resulting from $$\Gamma ^{i}_{ {0}{0}}{}$$ . Keeping only this c-independent term the geodesic equation reads
$$\displaystyle \begin{aligned} {d^2 x^i \over dt^2} = - c^2 \Gamma^{i}_{ {0}{0}}{} = U_{,i} \, . \end{aligned} $$
(5.3.6)
This, however, is precisely the equation of free-fall of a sufficiently small test body in Newton’s theory of gravity in Galilean coordinates (Cartesian and inertial).

5.4 Einstein’s Theory of Gravity

Einstein’s theory of gravity is the ‘simplest’ of all reasonable metric theories of gravity. In Einstein’s theory there are no other g-fields but only one space-time metric that also describes gravity.

Metric property 3 indicates an intimate relation between Newton’s theory of gravity and a relativistic one. In both theories test particles move along geodesics of the space-time geometry. As we have seen the Newtonian field equation for the potential U relates the Ricci-tensor of space-time with the field generating source. Now in relativity the source of the gravitational field obviously must be the energy-momentum tensor T μν and Einstein’s field equations for the metric tensor take the form
$$\displaystyle \begin{aligned} \mathcal{F}^{\mu\nu}(g,\partial g, \partial^2 g) = \kappa \, T^{\mu\nu} \end{aligned}$$
where $$\mathcal {F}^{\mu \nu }$$ is a function of g μν and its first and second partial derivatives with respect to the coordinates x μ. Because of the conservation laws for energy and momentum, Eq. (4.​6.​14) we have to require
$$\displaystyle \begin{aligned} {\mathcal{F}^{\mu\nu}}_{;\nu} = 0 \, . \end{aligned} $$
(5.4.1)

Theorem 5.1 (Lovelock 1972)

The most general tensor $$\mathcal {F}^{\mu \nu }(g,\partial g, \partial ^2 g)$$ that is divergenceless, i.e., obeys Eq.(5.4.1) is of the form
$$\displaystyle \begin{aligned} \mathcal{F}^{\mu\nu} = a G^{\mu\nu} + b g^{\mu\nu} \end{aligned} $$
(5.4.2)

where G μν are the components of the Einstein-tensor.

The usual Einstein’s field equations are obtained with a = 1 and b = 0:
$$\displaystyle \begin{aligned} G^{\mu\nu} = \kappa \, T^{\mu\nu} \, \end{aligned} $$
(5.4.3)
or
$$\displaystyle \begin{aligned} R^{\mu\nu} - \frac{1}{2} g^{\mu\nu} R = \kappa \, T^{\mu\nu} \, . \end{aligned} $$
(5.4.4)
Another important form of the field equations is obtained by contracting equation (5.4.4) with g μν (i.e., by taking its trace):
$$\displaystyle \begin{aligned} R - 2 R = - R = \kappa \, T \, , \end{aligned}$$
where
$$\displaystyle \begin{aligned} T \equiv g_{\mu\nu} T^{\mu\nu} \end{aligned} $$
(5.4.5)
is the trace of the energy-momentum tensor. Inserting this result for the curvature scalar R into Einstein’s field equations leads to the alternative form
$$\displaystyle \begin{aligned} R^{\mu\nu} = \kappa \, \left( T^{\mu\nu} - \frac{1}{2} g^{\mu\nu} T \right) \equiv \kappa \, \hat T^{\mu\nu} \, . \end{aligned} $$
(5.4.6)
Finally we have to determine the coupling constant κ. To this end we consider the ‘Newtonian limit’ of these field equations. In Newton’s theory only the matter density ρ acts as source of the gravitational field. This density to lowest order is contained in the time-time component of the energy-momentum tensor, considering a continuous distribution of energy and momentum. From (4.​6.​7) we see that
$$\displaystyle \begin{aligned} T^{00} = - T = \rho c^2 + \dots \end{aligned} $$
(5.4.7)
and for that reason the Newtonian field equation must be contained in the time-time component of (5.4.6):
$$\displaystyle \begin{aligned} R^{00} = \kappa \left(T^{00} - \frac{1}{2} g^{00} T \right) = \frac{1}{2} \kappa \rho c^2 \, . \end{aligned}$$
The left hand side to order c −2 can be taken from Eq. (3.​2.​21) keeping in mind that now x 0 = ct and the dimension of the Einstein tensor is (length)−2:
$$\displaystyle \begin{aligned} R_{00} = R^{00} = -{ \Delta U \over c^2} + \dots \, . \end{aligned}$$
Hence to lowest order the Einstein field equations lead to
$$\displaystyle \begin{aligned} \Delta U = - \frac{1}{2} \kappa \, \rho c^4 \end{aligned}$$
and a comparison with the Poisson equation (3.​2.​19) shows that
$$\displaystyle \begin{aligned} \kappa = {8 \pi \, G \over c^4} \, . \end{aligned}$$
Einstein’s field equations form a complicated set of ten partial differential equations of second order. Because of the Bianchi identities these ten equations are not independent from each other but only six of them. Hence, the equations determine six out of ten degrees of freedom of the metric tensor g μν. Four degrees of freedom for the metric tensor remain, expressing the freedom in the choice of the four space-time coordinates. Of course the field equations cannot tell what coordinates should be used; instead the coordinates can be fixed by four (more or less) arbitrary conditions for the metric tensor. This is the coordinate or “gauge” freedom of the theory. This gauge freedom is one of the most important differences to the classical Newtonian case. In Newton’s theory time is absolute, so there is a preferred time coordinate which is fixed uniquely up to origin and unit. Out of the many possible spatial coordinates the inertial (Cartesian) ones which in Newton’s theory exist globally are preferred. They are determined uniquely up to origin, unit and orientation in space (determined e.g., by three Euler angles). All these preferred coordinates, however, do not exist in Einstein’s theory of gravity. However, the situation is not too bad for isolated systems with an asymptotically flat space-time. E.g., the solar system might be idealized in this manner: we forget about distant masses and think of the solar system as being isolated. Then far from the solar system the gravitational field will become very small and space-time will approach flat space-time from Special Relativity Theory in this idealized picture. Then in the asymptotic region preferred (inertial and Cartesian) coordinates exist such that there
$$\displaystyle \begin{aligned} g_{\mu\nu} \rightarrow \eta_{\mu\nu} \, . \end{aligned} $$
(5.4.8)
If, however, we get closer to the gravitating masses preferred coordinates cease to exist; i.e., many different coordinates have equal rights.
If we choose a = 1 and b =  Λ in Lovelock’s Theorem then we end up with field equations of the form with a Λ term
$$\displaystyle \begin{aligned} R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu} \, . \end{aligned} $$
(5.4.9)
In this case the constant Λ is called the cosmological constant. It is obvious that the Λ-term can be absorbed in the energy-momentum tensor by replacing T μν by
$$\displaystyle \begin{aligned} \tilde T_{\mu\nu} \equiv T_{\mu\nu} - \kappa^{-1} \Lambda g_{\mu\nu} \, . \end{aligned} $$
(5.4.10)
Usually it is assumed that Λ is related with the energy density of the quantum vacuum pervading the whole universe and might have a value of about 10−52 m−2 (Peebles and Ratra 2003). Metric (5.4.9) plays an important role in modern cosmology.

5.5 The Problem of Observables

Since in Einstein’s theory of gravity the coordinates usually have no direct physical meaning the problem of observables is a serious one. It should be clear that observables are independent of any set of coordinates used by some theorist to describe the system of interest. In other words: observables have to be described by scalars, coordinate independent quantities. First one chooses some appropriate coordinate system and draws a coordinate picture of the system of interest. Then one constructs the observables as scalars from such a coordinate picture.

5.5.1 The Ranging Observable

Let us consider a typical astronomical measurement in the solar system: lunar laser ranging (LLR). Here laser pulses are emitted from LLR-stations on the Earth to retroreflectors on the lunar surface. A few photons per pulse find their way back into the receiving telescope of the station and one measures the total travel time of a pulse from the station to the Moon and back. This situation is depicted in Fig. 5.3. In the right part of the figure we see the world-line of the clock with the two events E: emission of the pulse and R: reception of the pulse. The observed time interval between E and R is then given by
$$\displaystyle \begin{aligned} \Delta \tau = \int_E^R d\tau \end{aligned} $$
(5.5.1)
with
$$\displaystyle \begin{aligned} d\tau^2 = - {1 \over c^2} ds^2 \, . \end{aligned}$$
In practise this indicated time interval Δτ can then be related with a corresponding interval of some other time scale.
../images/447007_1_En_5_Chapter/447007_1_En_5_Fig3_HTML.png
Fig. 5.3

Left: A central observable for celestial mechanics, the ranging observable, is a propagation time interval between emission and reception of some electromagnetic pulse. In Lunar Laser Ranging it is a laser pulse that travels from some LLR-station on the Earth to some retro-reflector on the lunar surface and back to the ground station. Right: the observable is the proper time interval that has elapsed between the instant of emission and the instant of reception of an electromagnetic pulse

5.5.2 The Spectroscopic Observable

We now consider the following problem: one observer emits some monochromatic electromagnetic wave of frequency f E. Another observer receives this signal and measures the frequency f R and we ask about the relation between the two frequencies. If we concentrate upon one single light-ray propagating from the emitter to the receiver the situation is shown in Fig. 5.4. Here γ E is the world-line of the emitter, γ R that of the receiver, γ ∗ that of the light ray. Let $$u^\mu _E$$ be the 4-velocity of the emitter at the point of emission, $$u^\mu _R$$ that of the receiver at the point of reception. Let k μ be the tangent vector onto γ ∗ then according to (4.​3.​7) the frequency ratio is given by
$$\displaystyle \begin{aligned} {f_{\mathrm{R}} \over f_{\mathrm{E}}} = {(g_{\mu\nu} k^\mu u^\nu)_{\mathrm{R}} \over (g_{\mu\nu} k^\mu u^\nu)_{\mathrm{E}}} \, . \end{aligned} $$
(5.5.2)
Let us analyze this situation in Minkowski space in the absence of gravitational fields. Let us choose a Minkowskian coordinate system such that the receiver is at rest in the event of reception, i.e.,
$$\displaystyle \begin{aligned} u_R^\mu = (c,\mathbf{0}) \, . \end{aligned}$$
If the emitter has coordinate velocity v at the point of emission then
$$\displaystyle \begin{aligned} u_E^\mu = \gamma (c,\mathbf{v}) \, . \end{aligned}$$
Since k μ is a null-vector we can write in Minkowskian coordinates
$$\displaystyle \begin{aligned} k^\mu = \mathrm{const.} \times (1,\mathbf{n}) \end{aligned} $$
(5.5.3)
with
$$\displaystyle \begin{aligned} \delta_{ij}\, n^i n^j = 1 \, . \end{aligned}$$
The normalization constant in k μ will not play a role if only frequency ratios are considered. With
$$\displaystyle \begin{aligned} \boldsymbol{\beta} \equiv {\mathbf{v} \over c} \end{aligned} $$
(5.5.4)
we then get
$$\displaystyle \begin{aligned} {f_{\mathrm{R}} \over f_{\mathrm{E}}} = { (-k^0 u^0 + k^i u^i)_{\mathrm{R}} \over (-k^0 u^0 + k^i u^i)_{\mathrm{ E}} } = \left[ \gamma \left( 1 - \boldsymbol{\beta} \cdot \mathbf{n} \right) \right]^{-1} \end{aligned}$$
or
$$\displaystyle \begin{aligned} f_{\mathrm{R}} = f_{\mathrm{E}}\, {\sqrt{ 1 - \boldsymbol{\beta}^2} \over 1 - \boldsymbol{\beta}\cdot \mathbf{n} } \, . \end{aligned} $$
(5.5.5)
This is the well-known formula for the Doppler-effect in electromagnetism.
../images/447007_1_En_5_Chapter/447007_1_En_5_Fig4_HTML.png
Fig. 5.4

The spectroscopic observable is the frequency ratio f R∕f E. Some observer (emitter) emits some electromagnetic signal of frequency f E. This signal is observed by another observer (receiver) who measures the frequency f R

5.5.3 The Astrometric Observable

In astrometry the principle observable is the observed angle between two incident light-rays. This situation is depicted in Fig. 5.5. Here γ(λ) is the worldline of the observer, $$\gamma ^*_1$$ and $$\gamma ^*_2$$ are two light-rays from two different astronomical sources that are simultaneously observed by the observer in some event O. Let u μ be the 4-velocity of the observer in O, $$k^\mu _1$$ and $$k^\mu _2$$ be the wave vectors of the two incident light-rays. Then
$$\displaystyle \begin{aligned} P^\mu_\nu \equiv \delta^\mu_\nu + {1 \over c^2} u^\mu u_\nu\end{aligned} $$
(5.5.6)
is a projection tensor that projects vectors into their components perpendicular to u μ, i.e.,
$$\displaystyle \begin{aligned} P^\mu_\nu \, u^\nu = u^\mu + {1 \over c^2} u^\mu u_\nu u^\nu = 0 \, \end{aligned} $$
(5.5.7)
since u ν u ν = −c 2. In some sense u μ points into the time-direction of the observer and the projection operator points into the space ‘experienced’ by the observer. Now $$k^\mu _i$$ are null-vectors but
$$\displaystyle \begin{aligned} {\overline k}^\mu = P^\mu_\nu k^\nu \end{aligned} $$
(5.5.8)
is a spacelike vector of non-vanishing length. For
$$\displaystyle \begin{aligned} u^\mu = \gamma c (1, \boldsymbol{\beta}) \; \qquad k^\mu = (1,\mathbf{n}) \end{aligned}$$
we find
$$\displaystyle \begin{aligned} u_\nu k^\nu = - \gamma c \,(1 - \boldsymbol{\beta} \cdot \mathbf{n}) \end{aligned} $$
(5.5.9)
and therefore
$$\displaystyle \begin{aligned} {\overline k}^\mu = k^\mu - \frac 1c \gamma \, (1 - \boldsymbol{\beta}\cdot \mathbf{n}) u^\mu \, . \end{aligned} $$
(5.5.10)
From this it is not difficult to see that
$$\displaystyle \begin{aligned} \vert{\overline k}^\mu\vert \equiv (g_{\mu\nu} {\overline k}^\mu {\overline k}^\nu)^{1/2} = \gamma ( 1 - \boldsymbol{\beta} \cdot \mathbf{n}) \, . \end{aligned} $$
(5.5.11)
The observed angle θ between two incident light-rays $$\gamma _1^*$$ and $$\gamma _2^*$$ is generally given by
$$\displaystyle \begin{aligned} \cos\theta = { g_{\mu\nu} {\overline k}_1^\mu {\overline k}_2^\nu \over \vert {\overline k}_1^\mu\vert \, \vert {\overline k}_2^\mu \vert } \, . \end{aligned} $$
(5.5.12)
In the absence of gravity fields from (5.5.12) we get
$$\displaystyle \begin{aligned} \cos\theta = { {\mathbf{n}}_1 \cdot {\mathbf{n}}_2 - 1 + \gamma^2(1 - \boldsymbol{\beta} \cdot {\mathbf{n}}_1) (1 - \boldsymbol{\beta} \cdot {\mathbf{n}}_2) \over \gamma^2 (1 - \boldsymbol{\beta} \cdot {\mathbf{n}}_1) (1 - \boldsymbol{\beta} \cdot {\mathbf{n}}_2) } \, . \end{aligned} $$
(5.5.13)
This is the aberration formula if gravity fields play no role. A Taylor expansion in terms of c −1 yields
$$\displaystyle \begin{aligned} \begin{array}{rcl} \cos\theta = &\displaystyle &\displaystyle {\mathbf{n}}_1 \cdot {\mathbf{n}}_2 + ({\mathbf{n}}_1 \cdot {\mathbf{n}}_2 - 1) \left[ ({\mathbf{n}}_1 + {\mathbf{n}}_2) \cdot \boldsymbol{\beta} + ({\mathbf{n}}_1 \cdot \boldsymbol{\beta})^2 \right. \\ &\displaystyle &\displaystyle +\, ({\mathbf{n}}_2 \cdot \boldsymbol{\beta})^2 + \left. ({\mathbf{n}}_1\cdot\boldsymbol{\beta})({\mathbf{n}}_2\cdot\boldsymbol{\beta}) - \boldsymbol{\beta}^2 \right] + \mathcal{O}(c^{-3}) \, . {} \end{array} \end{aligned} $$
(5.5.14)
../images/447007_1_En_5_Chapter/447007_1_En_5_Fig5_HTML.png
Fig. 5.5

The astrometric observable: the observed angle θ between two incident light-rays $$\gamma _1^*$$ and $$\gamma _2^*$$ . The observer’s worldline is γ(λ) and $$k^\mu _i$$ are tangent vectors to the light-rays

5.6 Tetrads and Tetrad Induced Coordinates

Consider some massless observer E that moves through empty space with a space capsule and wants to perform some local experiment inside of his spacecraft. Let us describe the motion of E in some coordinate system x μ by some timelike worldline $$\mathcal {L}_{\mathrm {E}}$$ , given by $$z^\mu _{\mathrm {E}}(\lambda )$$ , where λ is some affine parameter. Let us choose this parameter λ as the observer’s proper time τ also denoted by T. The tangent vector $$u^\mu \equiv dz^\mu _{\mathrm {E}}/dT$$ then is the observer’s 4-velocity that is normalized according to
$$\displaystyle \begin{aligned} g_{\mu\nu} u^\mu u^\nu = \frac {g_{\mu\nu} \, dz^\mu_{\mathrm{E}} dz^\nu_{\mathrm{ E}}}{dT \cdot dT} = \left( {ds \over dT} \right)^2 = - c^2 \, , \end{aligned} $$
(5.6.1)
since ds 2 = −c 2 dτ 2 along the observer’s world-line. In the following we will denote the unit vector in the direction of u μ by
$$\displaystyle \begin{aligned} e_{(0)}^\mu \equiv {1 \over c} u^\mu \, . \end{aligned} $$
(5.6.2)
Let
$$\displaystyle \begin{aligned} a^\mu \equiv u^\mu_{;\nu} u^\nu \end{aligned} $$
(5.6.3)
be the observer’s 4-acceleration, a vector that is perpendicular to u μ since
$$\displaystyle \begin{aligned} g_{\mu\nu} a^\mu u^\nu = {1 \over 2} (g_{\mu\nu} u^\mu u^\nu)_{;\sigma} u^\sigma = 0 \end{aligned} $$
(5.6.4)
in virtue of the normalization condition and g μν;σ = 0.
A set of four orthonormal vectors $$e^\mu _{(\alpha )} (\alpha = 0,1,2,3)$$ with $$e^\mu _{(0)}$$ being given by (5.6.2) and
$$\displaystyle \begin{aligned} g_{\mu\nu} e^\mu_{(\alpha)} e^\nu_{(\beta)} = \eta_{\alpha\beta} \end{aligned} $$
(5.6.5)
along $$\mathcal {L}_{\mathrm {E}}$$ is called a tetrad field along $$\mathcal {L}_{\mathrm {E}}$$ . Such tetrad fields are valuable quantities that can be used in different respects, e.g., for the construction of observables.
They can also be used to define useful local coordinates X α = (cT, X a) for the observer. First the local time coordinates T will be chosen as proper time τ of the observer whose world-line should be given by X a = 0, i.e., the observer is located at the spatial origin of his local coordinate system. Next we define: a local system of coordinates X α is called tetrad-induced if
$$\displaystyle \begin{aligned} e^\mu_{(\alpha)} = \left. {\partial x^u \over \partial X^\alpha} \right\vert_{\mathcal{L}_{\mathrm{E}}} \, . \end{aligned} $$
(5.6.6)
From this definition we find that the tetrad vectors in tetrad-induced coordinates (TIC) take a particularly simple form
$$\displaystyle \begin{aligned} E^\beta_{(\alpha)} = \left.{\partial X^\beta \over \partial x^\mu} \, e^\mu_{(\alpha)} \right\vert_{\mathcal{L}_{\mathrm{E}}} = \left. {\partial X^\beta \over \partial x^\mu} {\partial x^\mu \over \partial X^\alpha} \right\vert_{\mathcal{L}_{\mathrm{E}}} = \delta_{\alpha\beta} \, . \end{aligned} $$
(5.6.7)
Using this condition in TIC we find
$$\displaystyle \begin{aligned} G_{\alpha\beta} \vert_{\mathcal{L}_{\mathrm{E}}} = \eta_{\alpha\beta} \, . \end{aligned} $$
(5.6.8)
Hence, TIC are locally Minkowskian. We will now construct certain TIC in the neighbourhood of $$\mathcal {L}_{\mathrm {E}}$$ by imposing certain constraints on the Christoffel-symbols. To this end we consider the following quantities
$$\displaystyle \begin{aligned} ({\mathbf{E}}_{(\gamma)}, D_{(\alpha)} {\mathbf{E}}_{(\beta)}) \equiv G_{\rho\sigma} E^\rho_{(\beta);\kappa} E^\kappa_{(\alpha)} E^\sigma_{(\gamma)} \, . \end{aligned} $$
(5.6.9)
Because of the simple form of tetrad vectors in TIC, Eq. (5.6.7), we have
$$\displaystyle \begin{aligned} E^\rho_{(\beta);\kappa} = E^\rho_{(\beta),\kappa} + \Gamma^\rho_{\kappa\tau} E^\tau_{(\beta)} = \Gamma^\rho_{\kappa\tau} E^\tau_{(\beta)} \end{aligned}$$
that leads to
$$\displaystyle \begin{aligned} \left.({\mathbf{E}}_{(\gamma)}, D_{(\alpha)} {\mathbf{E}}_{(\beta)} \right\vert_{\mathcal{L}_{\mathrm{E}}} = \left. \eta_{\rho\gamma} \Gamma^\rho_{\alpha\beta} \right\vert_{\mathcal{L}_{\mathrm{E}}} \, . \end{aligned} $$
(5.6.10)

Lemma 5.1

The Christoffel-symbols in TIC obey the following relations at $$\mathcal {L}_{\mathrm { E}}$$ :
$$\displaystyle \begin{aligned} \begin{aligned} \Gamma^0_{00} &= 0 \, , \\ \Gamma^a_{00} &= \Gamma^0_{0a} = {1 \over c^2} A^a \, , \\ \Gamma^b_{0a} &= {1 \over c} \Omega_{(a)(b)} \, , \end{aligned} \end{aligned} $$
(5.6.11)
where A a are the spatial tetrad components of the 4-acceleration of E, i.e.,
$$\displaystyle \begin{aligned} A^a \equiv g_{\mu\nu}e^\mu_{(a)} a^\nu \end{aligned} $$
(5.6.12)
and
$$\displaystyle \begin{aligned} \Omega_{(a)(b)} \equiv c \cdot ({\mathbf{E}}_{(b)},D_{(0)} {\mathbf{E}}_{(a)}) \, . \end{aligned} $$
(5.6.13)

The quantities Ω ( a)( b) are called Ricci-rotation coefficients.

Exercise 5.1

Use the orthonormality of tetrad vectors to proof the antisymmetry of rotation coefficients
$$\displaystyle \begin{aligned} \Omega_{(a)(b)} = -\Omega_{(b)(a)} \, . \end{aligned} $$
(5.6.14)

The proof of Lemma 5.1 follows from (5.6.10), the definition of the 4-acceleration and the orthonormality of tetrad vectors. This Lemma implies that all Christoffel-symbols of TIC at the observer’s worldline are fixed apart from $$\Gamma ^\alpha _{bc}$$ . We now have several possibilities to fix these remaining quantities at $$\mathcal {L}_{\mathrm {E}}$$ .

Exercise 5.2

Suppose the X, Y, Z coordinates lines $$Y^\mu _{(\alpha )}$$ , which are integral curves to the tetrad $$e^\mu _{(\alpha )}$$ , are geodesics, parametrized with proper length s. Show that for that case
$$\displaystyle \begin{aligned} \left. \Gamma^\alpha_{bc} \right\vert_{\mathcal{L}_{\mathrm{E}}} = 0 \, . \end{aligned} $$
(5.6.15)
Corresponding TIC will be called local geodetic proper coordinates.
The last Exercise shows one possible choice for $$\Gamma ^\alpha _{bc}$$ . Another one is given by TIC that are locally harmonic. The harmonicity condition at $$\mathcal {L}_{\mathrm {E}}$$ can be written in the form
$$\displaystyle \begin{aligned} G^{\alpha\beta} \Gamma^\lambda_{\alpha\beta} = 0 \, ,\end{aligned} $$
i.e.,
$$\displaystyle \begin{aligned} \Gamma^\lambda_{aa} = \Gamma^\lambda_{00} \, .\end{aligned} $$
(5.6.16)
One solution of the harmonicity condition along $$\mathcal {L}_{\mathrm {E}}$$ reads
$$\displaystyle \begin{aligned} \begin{aligned} \Gamma^0_{bc} &= 0 \, , \\ \Gamma^a_{bc} &= - {1 \over c^2} (\delta_{ab} A^c + \delta_{ac} A^b - \delta_{bc} A^a) \, . \end{aligned} \end{aligned} $$
(5.6.17)
We will call local TIC with such Christoffel-symbols local harmonic proper coordinates.
Because the covariant derivative of the metric tensor vanishes, i.e.,
$$\displaystyle \begin{aligned} 0 = G_{\alpha\beta;\gamma} = G_{\alpha\beta,\gamma} - \Gamma^\delta_{\alpha\gamma} G_{\delta\beta} - \Gamma^\delta_{\beta\gamma} G_{\alpha\delta} \, , \end{aligned} $$
(5.6.18)
the Christoffel-symbols at $$\mathcal {L}_{\mathrm {E}}$$ determine the partial derivatives of G αβ at the worldline of E.

Exercise 5.3

Show that condition (5.6.18) for local geodetic proper coordinates proper coordinates leads to
$$\displaystyle \begin{aligned} G_{\alpha\beta,0} = 0 \, \quad G_{ab,c} = 0 \, \quad G_{00,a} = 2 A^a \, , \quad G_{0a,c} = {1 \over c} \epsilon_{abc} \Omega^b \, . \end{aligned} $$
(5.6.19)
Together with $$G_{\alpha \beta } \vert _{\mathcal {L}_{\mathrm {E}}} = \eta _{\alpha \beta }$$ this leads to a metric tensor of the form
$$\displaystyle \begin{aligned} \begin{aligned} G_{00} &= - \left(1 + {2 \over c^2} \mathbf{A} \cdot \mathbf{X}\right) + \mathcal{O}(\vert\mathbf{X}\vert^2) \\ G_{0a} &= {1 \over c} \epsilon_{abc} \Omega^b X^c + \mathcal{O}(\vert\mathbf{X}\vert^2) \\ G_{ab} &= \delta_{ab} + \mathcal{O}(\vert\mathbf{X}\vert^2) \end{aligned} \end{aligned} $$
(5.6.20)
with
$$\displaystyle \begin{aligned} \Omega_{(a)(b)} = \epsilon_{abc} \Omega^c \, . \end{aligned} $$
(5.6.21)
For local harmonic proper coordinates condition (5.6.18) leads to
$$\displaystyle \begin{aligned} G_{\alpha\beta,0} = 0 \, \quad G_{ab,c} = {1 \over c^2} \delta_{ab} A^c \, \quad G_{00,a} = 2 A^a \, , \quad G_{0a,c} = {1 \over c} \epsilon_{abc} \Omega^b \end{aligned} $$
(5.6.22)
at the observer’s world-line. Using $$G_{\alpha \beta } \vert _{\mathcal {L}_{\mathrm {E}}} = \eta _{\alpha \beta }$$ this leads to a metric tensor of the form
$$\displaystyle \begin{aligned} \begin{aligned} G_{00} &= - \left(1 + {2 \over c^2} \mathbf{A} \cdot \mathbf{X}\right) + \mathcal{O}(\vert\mathbf{X}\vert^2) \\ G_{0a} &= {1 \over c} \epsilon_{abc} \Omega^b X^c + \mathcal{O}(\vert\mathbf{X}\vert^2) \\ G_{ab} &= \delta_{ab}\left( 1 - {2 \over c^2} \mathbf{A} \cdot \mathbf{X} \right) + \mathcal{O}(\vert\mathbf{x}\vert^2) \, , \end{aligned} \end{aligned} $$
(5.6.23)
where Ωb is again given by relation (5.6.21).
Finally, let us try to understand the meaning of our ‘angular velocity’ Ωb. To this end the definition of the Fermi-derivative is useful. Let B μ be some contravariant vector-field along $$\mathcal {L}_{\mathrm {E}}$$ with tangent vector field e μ = u μ∕c and 4-acceleration a μ. Then the Fermi-derivative D F of B μ is defined by
$$\displaystyle \begin{aligned} D_{\mathrm{F}} B^\mu \equiv D_u B^\mu + {1 \over c^3} (u_\nu B^\nu)a^\mu - {1 \over c^3} (a_\nu B^\nu) u^\mu \end{aligned} $$
(5.6.24)
with
$$\displaystyle \begin{aligned} D_u B^\mu \equiv {1 \over c} {B^\mu}_{;\nu} u^\nu = {B^\mu}_{;\nu} e^\mu_{(0)} \, . \end{aligned} $$
(5.6.25)

Exercise 5.4

Show that the Fermi-derivative has the following properties:

  1. (i)

    $$D_{\mathrm {F}} e^\mu _{(0)} = 0$$ .

     
  2. (ii)
    Let A μ and B μ be two contravariant vector-fields along $$\mathcal {L}_{\mathrm {E}}$$ with
    $$\displaystyle \begin{aligned} D_{\mathrm{F}} A^\mu = D_{\mathrm{F}} B^\mu = 0 \, . \end{aligned}$$
    Then
    $$\displaystyle \begin{aligned} g_{\mu\nu} A^\mu B^\nu \vert_{\mathcal{L}_{\mathrm{E}}} = \mathrm{const.} \end{aligned}$$
     
  3. (iii)
    Let A μ be some contravariant vector-field along $$\mathcal {L}_{\mathrm {E}}$$ , perpendicular to u μ, then
    $$\displaystyle \begin{aligned} D_{\mathrm{F}} A^\mu = (D_u A^\mu)_\perp \, , \end{aligned}$$
    where ⊥ denotes the projection of a vector B μ perpendicular to u μ, i.e.,
    $$\displaystyle \begin{aligned} B^\mu_\perp \equiv \left( \delta^\mu_\nu + {1 \over c^2} u^\mu u_\nu \right) B^\nu \, . \end{aligned}$$
     
Let us now consider the vector-field
$$\displaystyle \begin{aligned} C^\mu \equiv D_u e^\mu_{(a)} \vert_{\mathcal{L}_{\mathrm{E}}} \, . \end{aligned}$$
Obviously we can decompose C μ according to
$$\displaystyle \begin{aligned} C^\mu = - \left[ C_\sigma e^\sigma_{(0)} \right] \, e^\mu_{(0)} + \left[ C_\sigma e^\sigma_{(b)} \right] \, e^\mu_{(b)} \end{aligned}$$
i.e., at the observer’s worldline we get
$$\displaystyle \begin{aligned} D_u e^\mu_{(a)} = - g_{\rho\sigma}(D_u e^\rho_{(a)}) \, e^\sigma_{(0)} e^\mu_{(0)} + g_{\rho\sigma} (D_u e^\rho_{(a)}) \, e^\sigma_{(b)} e^\mu_{(b)} \, . \end{aligned} $$
(5.6.26)
From the orthonormality condition
$$\displaystyle \begin{aligned} g_{\rho\sigma} e^\sigma_{(0)} e^\rho_{(a)} = 0 \end{aligned}$$
we get
$$\displaystyle \begin{aligned} g_{\rho\sigma} (D_u e^\rho_{(a)}) e^\sigma_{(0)} + g_{\rho\sigma} (D_u e^\sigma_{(0)}) e^\rho_{(a)} = 0 \end{aligned}$$
that we can use to rewrite the first term in the right-hand side of (5.6.26). Adding to this equation a vanishing u μ a μ-term we get along $$\mathcal {L}_{\mathrm {E}}$$ :
$$\displaystyle \begin{aligned} D_{\mathrm{F}} e^\mu_{(a)} = {1 \over c} \Omega_{(a)(b)} e^\mu_{(b)} \, . \end{aligned} $$
(5.6.27)

This relation proofs the following:

Theorem 5.2

If the tetrad $$e^\mu _{(\alpha )}$$ along $$\mathcal {L}_{\mathrm {E}}$$ is Fermi-Walker transported, i.e.,
$$\displaystyle \begin{aligned} D_{\mathrm{F}} e^\mu_{(\alpha)} = 0 \, , \end{aligned}$$

the Ricci rotation-coefficients vanish, i.e., Ω (a)(b) = 0.

Finally let us study the motion of a test-body in free-fall in the vicinity of the observer. Let $$Z^\alpha _{\mathrm {B}} (T) \equiv (cT, Z^a)$$ denote the world-line of this test-body, given by a geodesic of the form
$$\displaystyle \begin{aligned} {d^2 Z^a \over dT^2} = - \Gamma^a_{\alpha\beta} {dZ^\alpha \over dT} {dZ^\beta \over dT} + {1 \over c} \Gamma^0_{\alpha\beta}{dZ^\alpha \over dT}{dZ^\beta \over dT}{dZ^a \over dT} \, , \end{aligned}$$
that we will analyze at $$\mathcal {L}_{\mathrm {E}}$$ , i.e., for Z a = 0. By taking into account of the corresponding Christoffel-symbols at X a = 0 this equation in local harmonic proper coordinates takes the form
$$\displaystyle \begin{aligned} {d^2 \mathbf{Z} \over dT^2} + 2 (\boldsymbol{\Omega} \times \mathbf{V}) = - \left( 1 - {{\mathbf{V}}^2 \over c^2} \right) \, \mathbf{A} \end{aligned} $$
(5.6.28)
with V ≡ d Z∕dT. Hence, Ω describes nothing but a Coriolis-force due to the rotational motion of spatial axes. The term on the right-hand side of (5.6.28) presents the inertial acceleration due to the 4-acceleration of the observer.

Exercise 5.5

Show that in local geodetic proper coordinates the geodesic equation at $$\mathcal {L}_{\mathrm {E}}$$ takes the form (see also Misner et al. 1973, Exercise (13.14))
$$\displaystyle \begin{aligned} {d^2 \mathbf{Z} \over dT^2} + 2 (\boldsymbol{\Omega} \times \mathbf{V}) = - \mathbf{A} + {2 \over c^2} \mathbf{V} (\mathbf{V} \cdot \mathbf{A}) \, . \end{aligned} $$
(5.6.29)

Local TIC will be called dynamically non-rotating or locally inertial if Ω = 0. In that case the local reference system will show no inertial forces due to the rotational motion of spatial basis vectors. Technically speaking this means that G 0a = 0 for dynamically non-rotating local coordinates. As we have seen the dynamically non-rotating local proper coordinates result from Fermi-transported tetrad vectors.

5.7 Proper Reference Systems of Accelerated Observers

Let us start with inertial Minkowskian coordinates x μ = (ct, x) and consider an observer that is moving along the x-axis with constant 4-acceleration, i.e.,
$$\displaystyle \begin{aligned} a^\mu a_\mu = - a^0 a^0 + a^1 a^1 = g^2 \, . \end{aligned} $$
(5.7.1)
Together with u μ u μ = −c 2 or
$$\displaystyle \begin{aligned} u^0 u^0 - u^1 u^1 = c^2 \end{aligned} $$
(5.7.2)
and u μ a μ = 0, i.e.,
$$\displaystyle \begin{aligned} u^0 a^0 - u^1 a^1 = 0 \end{aligned} $$
(5.7.3)
we get
$$\displaystyle \begin{aligned} g^2 = {a^1 a^1 \over u^0 u^0} c^2 = {a^0 a^0 \over u^1 u^1} c^2 \end{aligned}$$
or
$$\displaystyle \begin{aligned} g = c {a^1 \over u^0} = c {a^0 \over u^1} \, . \end{aligned} $$
(5.7.4)
Thus,
$$\displaystyle \begin{aligned} \begin{aligned} a^0 &= {d u^0 \over d\tau} = {g \over c} u^1 \\ a^1 &= {d u^1 \over d\tau} = {g \over c} u^0 \, . \end{aligned} \end{aligned} $$
(5.7.5)
A special solution of these two differential equations is given by
$$\displaystyle \begin{aligned} \begin{aligned} z_{\mathrm{obs}}^0(\tau) &= {c^2 \over g} \sinh \alpha \\ z_{\mathrm{obs}}^1(\tau) &= {c^2 \over g} \cosh \alpha \end{aligned} \end{aligned} $$
(5.7.6)
with
$$\displaystyle \begin{aligned} \alpha = {g \tau \over c} \, . \end{aligned}$$
From this we get
$$\displaystyle \begin{aligned} \begin{aligned} u^0(\tau) &= {dz_{\mathrm{obs}}^0 \over d\tau} = c \cosh \alpha \\ u^1(\tau) &= {dz_{\mathrm{obs}}^1 \over d\tau} = c \sinh \alpha \end{aligned} \end{aligned} $$
(5.7.7)
and
$$\displaystyle \begin{aligned} \begin{aligned} a^0(\tau) &= {du^0\over d\tau} = g \sinh \alpha \\ a^1(\tau) &= {du^1 \over d\tau} = g \cosh \alpha \, . \end{aligned} \end{aligned} $$
(5.7.8)
Since $$\cosh ^2 x - \sinh ^2 x = 1$$ the trajectory of the observer, $$\mathcal L_{\mathrm {obs}}$$ , is given by
$$\displaystyle \begin{aligned} x_{\mathrm{obs}}^2 - c^2 t_{\mathrm{obs}}^2 = {c^2 \over g^2} \, , \end{aligned} $$
(5.7.9)
i.e., by a hyperbola in our inertial Minkowskian coordinates. Next we construct a local co-moving tetrad field along $$\mathcal L_{\mathrm {obs}}$$ . The observer’s 4-velocity reads
$$\displaystyle \begin{aligned} u^\mu = {dz_{\mathrm{obs}} \over d\tau} = c (\cosh \alpha, \sinh \alpha, 0,0) \end{aligned}$$
so that the unit vector $$e_{(0)}^\mu $$ in the direction of u μ is given by
$$\displaystyle \begin{aligned} e_{(0)}^\mu = {1 \over c} u^\mu = (\cosh \alpha, \sinh \alpha, 0,0) \, . \end{aligned} $$
(5.7.10)
The corresponding spatial tetrad vectors, kinematically non-rotating with respect to the original Minkowskian coordinates, can then be chosen according to
$$\displaystyle \begin{aligned} \begin{aligned} e_{(1)}^\mu &= (\sinh \alpha, \cosh \alpha, 0,0) \\ e_{(2)}^\mu &= (0,0,1,0) \\ e_{(3)}^\mu &= (0,0,0,1) \, . \end{aligned} \end{aligned} $$
(5.7.11)
It is interesting to note that this tetrad field can easily be obtained from the Minkowskian basic vectors at rest:
$$\displaystyle \begin{aligned} \bar e_{(\alpha)}^\mu = \delta_{\alpha\mu} \, . \end{aligned} $$
(5.7.12)
We first write the tetrads $$e_{(\alpha )}^\mu $$ in terms of the observer’s coordinate velocity
$$\displaystyle \begin{aligned} v = {dz_{\mathrm{obs}} \over dt} = {dz_{\mathrm{obs}} \over d\tau} \left( {dt \over d\tau} \right)^{-1} = c \cdot \tanh \alpha \, . \end{aligned} $$
(5.7.13)
With
$$\displaystyle \begin{aligned} \beta \equiv {v \over c} = \tanh \alpha \, ; \qquad \gamma \equiv (1 - \beta^2)^{-1/2} = \cosh \alpha \end{aligned}$$
we get
$$\displaystyle \begin{aligned} \begin{aligned} e_{(0)}^\mu &= \gamma(1,\beta, 0,0) \\ e_{(1)}^\mu &= \gamma(\beta,1,0,0) \, . \end{aligned} \end{aligned} $$
(5.7.14)
From this we see that the co-moving tetrads can be obtained from $$\bar e_{(\beta )}^\mu $$ by means of a Lorentz-boost:
$$\displaystyle \begin{aligned} e^\mu_{(\alpha)} = \Lambda_{(\alpha)}^{(\beta)} \bar e_{(\beta)}^\mu \end{aligned} $$
(5.7.15)
with
$$\displaystyle \begin{aligned} \Lambda_{(\alpha)}^{(\beta)} = \left( \begin{array}{llll} \gamma & \gamma \beta & 0 & 0 \\ \gamma\beta & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right) \, . \end{aligned} $$
(5.7.16)
Nest we consider the coordinate transformation from inertial Minkowskian coordinates x μ = (ct, x) to local co-moving coordinates X α = (cT, X) with T = τ, the proper-time of the observer, with the ansatz
$$\displaystyle \begin{aligned} x^\mu(X^\alpha) = z^\mu(T) + e^\mu_{(a)} X^a + \xi^\mu(T,\mathbf{X}) \, , \end{aligned} $$
(5.7.17)
where ξ μ is at least of second order in |X|. For the Jacobian of this transformation
$$\displaystyle \begin{aligned} A^\mu_\nu \equiv {\partial x^\mu \over \partial X^\nu} \end{aligned} $$
(5.7.18)
we get
$$\displaystyle \begin{aligned} \begin{aligned} A_0^\mu &= e^\mu_{(0)} + {1 \over c} {d \over dT} e^\mu_{(a)} X^a + \xi^\mu_{,0} \\ A_a^\mu &= e^\mu_{(a)} + \xi^\mu_{,a} \, . \end{aligned} \end{aligned} $$
(5.7.19)
Since
$$\displaystyle \begin{aligned} {d \over dT} e_{(1)}^0 = {g \over c} e^0_{(0)} \, ; \quad {d \over dT} e_{(1)}^1 = {g \over c} e^1_{(0)} \end{aligned}$$
$$A_0^\mu $$ can be written in the form
$$\displaystyle \begin{aligned} A_0^\mu = \Phi \, e^\mu_{(0)} + \xi^\mu_{,0} \end{aligned} $$
(5.7.20)
with
$$\displaystyle \begin{aligned} \Phi = 1 + {g X \over c^2} \, . \end{aligned}$$
Now, our original Minkowskian coordinates are both geodetic and harmonic. For the local coordinates the condition of TIC ensures that the local metric tensor, G αβ, is Minkowskian at the origin, i.e., G αβ(X = 0) = η αβ. Higher order terms in |X|, linear, quadratic and higher are not fixed so far. We can fix them by coordinate conditions that we can impose on the local coordinates or we can specify the transformation functions ξ μ.
Let us start with
$$\displaystyle \begin{aligned} \xi_{\mathrm{G}}^\mu = 0 \, . \end{aligned} $$
(5.7.21)
Then the metric tensor G αβ in local coordinates according to the tensor transformation rule takes the form:
$$\displaystyle \begin{aligned} \begin{aligned} G_{00} &= A_0^\alpha A_0^\beta \eta_{\alpha\beta} = - A_0^0 A_0^0 + A_0^i A_0^i = - \Phi^2 \\ G_{0a} &= A_0^\alpha A_a^\beta \eta_{\alpha\beta} = - A_0^0 A_a^0 + A_0^i A_a^i = 0 \\ G_{ab} &= A_a^\alpha A_b^\beta \eta_{\alpha\beta} = - A_a^0 A_b^0 + A_a^i A_b^i = \delta_{ab} \\ \end{aligned} \end{aligned} $$
(5.7.22)
or
$$\displaystyle \begin{aligned} ds^2 = - \left( 1 + {gX \over c^2} \right)^2 c^2 dT^2 + d{\mathbf{X}}^2 \, . \end{aligned} $$
(5.7.23)

Exercise 5.6

Proof that the spatial coordinate lines X = X(λ);T = τ = const. are geodesics, i.e., the coordinates (cT, X) defined by $$\xi _{\mathrm { G}}^\mu = 0$$ are geodesic proper coordinates.

Next we will assume a special form of the local metric tensor
$$\displaystyle \begin{aligned} \begin{aligned} G_{00} &= - \left( 1 + {2 gX \over c^2} \right) + \mathcal{O}(c^{-4}) \\ G_{0a} &= 0 \\ G_{ab} &= \delta_{ab} \left( 1 - {2 g X \over c^2} \right) + \mathcal{O}(c^{-4}) \, . \end{aligned} \end{aligned} $$
(5.7.24)
In that case
$$\displaystyle \begin{aligned} G \equiv - \det (G_{\alpha\beta}) = 1 - {4 g X \over c^2} + \mathcal{O}(c^{-4}) \end{aligned} $$
(5.7.25)
and
$$\displaystyle \begin{aligned} \sqrt{G} G^{ab} = \delta_{ab} + \mathcal{O}(c^{-4}) \, , \end{aligned} $$
(5.7.26)
so that these spatial coordinates are harmonic up to terms of order c −4.

Exercise 5.7

Show that the local metric (5.7.24) can be obtained with
$$\displaystyle \begin{aligned} \begin{aligned} \xi^0_{\mathrm{H}}(T,\mathbf{X}) &= \mathcal{O}(c^{-3}) \\ \xi^a_{\mathrm{H}}(T,\mathbf{X}) &= {1 \over c^2} e^i_{(a)}(T) \left[ {1 \over 2} A_a {\mathbf{X}}^2 - X^a (\mathbf{A} \cdot \mathbf{X}) \right] + \mathcal{O}(c^{-4}) \end{aligned} \end{aligned} $$
(5.7.27)
where
$$\displaystyle \begin{aligned} A_a = \eta_{\mu\nu} e^\mu_{(a)} {d^2 z_{\mathrm{obs}}^\nu \over dT^2} = (g,0,0) \, . \end{aligned} $$
(5.7.28)

5.8 The Landau-Lifshitz Formulation of GR

5.8.1 The Landau-Lifshitz Field Equations

Landau and Lifshitz (1941, 1971) have derived a special form of the Einstein field equations, which presents a very useful starting point for solving the field equations with perturbative expansions. The atomic variable of the Landau-Lifshitz (LL) formalism is called h αβ, defined by (5.8.13) and the field equation in harmonic gauge are quasi-linear hyperbolic differential equations of the form ../images/447007_1_En_5_Chapter/447007_1_En_5_IEq62_HTML.gif, where ../images/447007_1_En_5_Chapter/447007_1_En_5_IEq63_HTML.gif is the flat space d’Alembertian (the flat space wave operator) and τ αβ is the gravitational source tensor, that itself contains h αβ-terms. Under a condition of ‘no incoming gravitational radiation’, the field equations (in harmonic gauge) can formally be solved in terms of retarded integrals (see (5.8.36) below) over quantities involving the atomic variable itself. To derive explicit results for h αβ, the source term τ αβ can be expanded in terms of small quantities as measures of weak gravitational fields, small velocities and small internal stresses. The MPM-formalism discussed in Chap. 7 presents such a scheme, where suitable expansions of τ αβ lead to fully explicit expressions for h αβ, even at high orders in the small parameters.

The LL-formalism is based upon the ‘gothic metric’, defined by
$$\displaystyle \begin{aligned} {\mathfrak{g}}^{\alpha\beta} \equiv \sqrt{-g} g^{\alpha\beta} \, , \end{aligned} $$
(5.8.1)
where $$g \equiv \det (g_{\alpha \beta })$$ . Note, that $${\mathfrak {g}}^{\alpha \beta }$$ is not a tensor but a tensor density. Let $${\mathfrak {g}}_{\alpha \beta }$$ be the inverse of $${\mathfrak {g}}^{\alpha \beta }$$ and $${\mathfrak {g}} \equiv \mathrm {det} ({\mathfrak {g}}^{\alpha \beta }) $$ . Then,
$$\displaystyle \begin{aligned} {\mathfrak{g}} = \det({\mathfrak{g}}^{\alpha\beta}) = \det( \sqrt{-g} g^{\alpha\beta}) = g^2 \det(g^{\alpha\beta}) = g \, . \end{aligned} $$
(5.8.2)
If we take the inverse matrix of
$$\displaystyle \begin{aligned} g^{\alpha\beta} = (-g)^{-1/2} {\mathfrak{g}}^{\alpha\beta} \end{aligned}$$
we therefore get
$$\displaystyle \begin{aligned} g_{\alpha\beta} = \sqrt{-g}\, {\mathfrak{g}}_{\alpha\beta} = \sqrt{-{\mathfrak{g}}}\, {\mathfrak{g}}_{\alpha\beta} \, . \end{aligned} $$
(5.8.3)
Let us define (e.g., Poisson and Will 2014)
$$\displaystyle \begin{aligned} H^{\alpha\mu\beta\nu} \equiv {\mathfrak{g}}^{\alpha\beta} {\mathfrak{g}}^{\mu\nu} - {\mathfrak{g}}^{\alpha\nu} {\mathfrak{g}}^{\beta\mu} \, . \end{aligned} $$
(5.8.4)
Now, H αμβν has the same properties as the Riemann tensor,
$$\displaystyle \begin{aligned} H^{\alpha\mu\beta\nu} = - H^{\mu\alpha\beta\nu} \, , \quad H^{\alpha\mu\beta\nu} = - H^{\mu\alpha\nu\beta} \, \quad H^{\alpha\mu\beta\nu} = + H^{\beta\nu\alpha\mu} \, . \end{aligned} $$
(5.8.5)
$$\displaystyle \begin{aligned} \partial_{\mu\nu} H^{\alpha\mu\beta\nu} = 2 \vert g \vert G^{\alpha\beta} + {16 \pi G \over c^4} \vert g \vert t^{\alpha\beta}_{\mathrm{LL}} \end{aligned} $$
(5.8.6)
where G αβ = R αβ − (1∕2)g αβ R is the Einstein tensor and $$t^{\alpha \beta }_{\mathrm {LL}}$$ is the Landau-Lifshitz pseudotensor:
$$\displaystyle \begin{aligned} \begin{aligned} {16 \pi G \over c^4} \vert g \vert t^{\alpha\beta}_{\mathrm{LL}} =& \ {\mathfrak{g}}^{\alpha\beta}_{,\lambda} {\mathfrak{g}}^{\lambda\mu}_{,\mu} - {\mathfrak{g}}^{\alpha\lambda}_{,\lambda}{\mathfrak{g}}^{\beta\mu}_{,\mu} + {1 \over 2} g^{\alpha\beta} g_{\lambda\mu} {\mathfrak{g}}^{\lambda\nu}_{,\rho} {\mathfrak{g}}^{\rho\mu}_{,\nu} \\ &- (g^{\alpha\lambda} g_{\mu\nu}{\mathfrak{g}}^{\beta\nu}_{,\rho} {\mathfrak{g}}^{\mu\rho}_{,\lambda} + g^{\beta\lambda}g_{\mu\nu} {\mathfrak{g}}^{\alpha\nu}_{, \rho} {\mathfrak{g}}^{\mu\rho}_{,\lambda}) + g_{\lambda\mu}g^{\nu\rho} {\mathfrak{g}}^{\alpha\lambda}_{,\nu} {\mathfrak{g}}^{\beta\mu}_{,\rho} \\ & + {1 \over 8} (2 g^{\alpha\lambda} g^{\beta\mu} - g^{\alpha\beta} g^{\lambda\mu})(2 g_{\nu\rho} g_{\sigma\tau} - g_{\rho\sigma} g_{\nu\tau}) {\mathfrak{g}}^{\nu\tau}_{,\lambda} {\mathfrak{g}}^{\rho\sigma}_{,\mu}. \end{aligned} \end{aligned} $$
(5.8.7)
Using the Einstein field equations for G αβ in (5.8.6) we get the exact field equations in the form
$$\displaystyle \begin{aligned} \partial_{\mu\nu} H^{\alpha\mu\beta\nu} = {16 \pi G \over c^4} \vert g \vert (T^{\alpha\beta} + t^{\alpha\beta}_{\mathrm{LL}}) \, . \end{aligned} $$
(5.8.8)
From the symmetry relations (5.8.5) we infer
$$\displaystyle \begin{aligned} \partial_{\beta\mu\nu} H^{\alpha\mu\beta\nu} = 0 \end{aligned} $$
(5.8.9)
or
$$\displaystyle \begin{aligned} \Theta^{\alpha\beta}_{,\beta} = 0 \, , \end{aligned} $$
(5.8.10)
where the Landau-Lifshitz complex Θαβ is given by
$$\displaystyle \begin{aligned} \Theta^{\alpha\beta} \equiv [ \vert g \vert (T^{\alpha\beta} + t^{\alpha\beta}_{\mathrm{ LL}})] \, . \end{aligned} $$
(5.8.11)
Equation (5.8.10) is equivalent to the local equations of motion: T αβ ;β = 0 .

Exercise 5.8

At a certain point $$p \in \mathcal {M}$$ choose local coordinates such that the first derivatives of the usual covariant metric tensor vanishes at p, i.e., g αβ,γ(p) = 0. Show that in such coordinates at p: $$\vert g \vert t^{\alpha \beta }_{\mathrm {LL}} = 0$$ and
$$\displaystyle \begin{aligned} \partial_{\mu\nu} H^{\alpha\mu\beta\nu} = 2 \vert g \vert G^{\alpha\beta} \end{aligned}$$
in accordance with (5.8.6). Details can be found in Poisson & Will (2014).

5.8.2 Harmonic Gauge

The formulation of the harmonic gauge is especially simple in terms of the gothic metric:
$$\displaystyle \begin{aligned} \partial_\beta {\mathfrak{g}}^{\alpha\beta} = 0 \qquad (\text{harmonic}\ \text{gauge}) \, . \end{aligned} $$
(5.8.12)
Let us define
$$\displaystyle \begin{aligned} h^{\alpha\beta} \equiv {\mathfrak{g}}^{\alpha\beta} - \eta^{\alpha\beta} \, . \end{aligned} $$
(5.8.13)
Then the harmonic condition reads:
$$\displaystyle \begin{aligned} \partial_\beta h^{\alpha\beta} = 0 \, . \end{aligned} $$
(5.8.14)

Exercise 5.9

Show that the following relations hold:
$$\displaystyle \begin{aligned} \begin{array}{rcl} (-g) &\displaystyle =&\displaystyle 1 + h + \frac{1}{2} h^2 - \frac{1}{2} h^{\alpha\beta} h_{\alpha\beta} + \mathcal{O}(G^3) \end{array} \end{aligned} $$
(5.8.15)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \sqrt{-g} &\displaystyle =&\displaystyle 1 + \frac{1}{2} h + \frac{1}{8} h^2 - \frac{1}{4} h^{\alpha\beta} h_{\alpha\beta} + \mathcal{O}(G^3) {} \vspace{-12pt}\end{array} \end{aligned} $$
(5.8.16)
$$\displaystyle \begin{aligned} \begin{array}{rcl} g_{\alpha\beta} &\displaystyle =&\displaystyle \eta_{\alpha\beta} - h_{\alpha\beta} + \frac{1}{2} h \,\eta_{\alpha\beta} + h_{\alpha\mu} h^\mu_\beta - \frac{1}{2} h h_{\alpha\beta} \\ &\displaystyle &\displaystyle + \left( \frac{1}{8} h^2 - \frac{1}{4} h^{\mu\nu} h_{\mu\nu} \right) \eta_{\alpha\beta} + \mathcal{O}(G^3) \, , \end{array} \end{aligned} $$
(5.8.17)
$$\displaystyle \begin{aligned} \begin{array}{rcl} g^{\alpha\beta} &\displaystyle =&\displaystyle \eta^{\alpha\beta} + h^{\alpha\beta} - \frac{1}{2} h \, \eta^{\alpha\beta} - \frac{1}{2} h h^{\alpha\beta}_\beta \\ &\displaystyle &\displaystyle + \left( \frac{1}{8} h^2 + \frac{1}{4} h^{\mu\nu} h_{\mu\nu} \right) \eta^{\alpha\beta} + \mathcal{O}(G^3) \, , {} \end{array} \end{aligned} $$
(5.8.18)
where indices on h αβ are lowered and contracted with the Minkowski metric η μν, thus h αβ ≡ η αμ η βν h μν and h ≡ η αβ h αβ.

Solution

From the definitions we have
$$\displaystyle \begin{aligned} h = \eta_{\alpha\beta} h^{\alpha\beta} = - h^{00} + h^{kk} \, ; \qquad h_{\alpha\beta} h^{\alpha\beta} = h^{00} - 2 h^{i0} h^{i0} + h^{ij} h^{ij} \, . \end{aligned} $$
(5.8.19)
Next we compute $$ - {\mathfrak {g}}$$ which is a polynomial of 4th order in G. We get
$$\displaystyle \begin{aligned} - {\mathfrak{g}} = - \mathrm{det} ({\mathfrak{g}}^{\alpha\beta}) = - \mathrm{det}(\eta^{\alpha\beta} + h^{\alpha\beta}) = 1 + h + \frac{1}{2} h^2 - \frac{1}{2} h^{\alpha\beta} h_{\alpha\beta} + \mathcal{O}(G^3) = -g \end{aligned}$$
and with (1 + x)−1∕2 = 1 − x∕2 + 3x 2∕8 + …, relation (5.8.16) follows. g αβ can then be obtained from g αβ = (−g)−1∕2(η αβ + h αβ) and it is easy to check that g αβ is inverse to g αβ, i.e., $$g^{\alpha \beta } g_{\beta \gamma } = \delta _{\alpha \gamma } + \mathcal {O}(G^3)$$ .

Exercise 5.10

Assume that h 00 is of order c −2, h 0i of order c −3 and h ij of order c −4. Show that (Poisson and Will 2014):
$$\displaystyle \begin{aligned} \begin{array}{rcl} g_{00} &\displaystyle =&\displaystyle -1 - \frac{1}{2} h^{00} - \frac{3}{8} (h^{00})^2 - \frac 5{16} (h^{00})^3 - \frac{1}{2} h^{kk} \left(1 + \frac{1}{2} h^{00}\right) \end{array} \end{aligned} $$
(5.8.20)
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle + \frac{1}{2} h^{0i}h^{0i} + \mathcal{O}_8 \end{array} \end{aligned} $$
(5.8.21)
$$\displaystyle \begin{aligned} \begin{array}{rcl} g_{0i} &\displaystyle =&\displaystyle h^{0i}\left(1 + \frac{1}{2} h^{00} \right) + \mathcal{O}_7 \end{array} \end{aligned} $$
(5.8.22)
$$\displaystyle \begin{aligned} \begin{array}{rcl} g_{ij} &\displaystyle =&\displaystyle \delta_{ij} \left[ 1 - \frac{1}{2} h^{00} - \frac{1}{8} (h^{00})^2 \right] - h^{ij} + \frac{1}{2} \delta_{ij} h^{kk} + \mathcal{O}_6 \end{array} \end{aligned} $$
(5.8.23)
$$\displaystyle \begin{aligned} \begin{array}{rcl} (-g) &\displaystyle =&\displaystyle 1 - h^{00} + h^{kk} + \mathcal{O}_6 \, . \end{array} \end{aligned} $$
(5.8.24)
In the harmonic gauge we have
../images/447007_1_En_5_Chapter/447007_1_En_5_Equ116_HTML.png
(5.8.25)
where
../images/447007_1_En_5_Chapter/447007_1_En_5_Equ117_HTML.png
(5.8.26)
is the flat-space d’Alembertian. Then the field equations take the form
../images/447007_1_En_5_Chapter/447007_1_En_5_Equ118_HTML.png
(5.8.27)
or
../images/447007_1_En_5_Chapter/447007_1_En_5_Equ119_HTML.png
(5.8.28)
with
$$\displaystyle \begin{aligned} \tau^{\alpha\beta} = \vert g \vert (T^{\alpha\beta} + t^{\alpha\beta}_{\mathrm{ LL}} + t^{\alpha\beta}_{\mathrm{H}}) \end{aligned} $$
(5.8.29)
and
$$\displaystyle \begin{aligned} \vert g \vert t^{\alpha\beta}_{\mathrm{H}} = {c^4 \over 16 \pi G} (\partial_\mu h^{\alpha\nu} \partial_\nu h^{\beta\mu} - h^{\mu\nu} \partial_{\mu\nu} h^{\alpha\beta}) \, . \end{aligned} $$
(5.8.30)
Usually the Einstein field equations in harmonic gauge are written in the form
../images/447007_1_En_5_Chapter/447007_1_En_5_Equ122_HTML.png
(5.8.31)
where the ‘gravitational source term’ Λαβ reads
$$\displaystyle \begin{aligned} \Lambda^{\alpha\beta} \equiv {16 \pi G \over c^4} \vert g \vert t^{\alpha\beta}_{\mathrm{LL}} + h^{\alpha\nu}_{,\mu} h^{\beta\mu}_{,\nu} - h^{\mu\nu} h^{\alpha\beta}_{,\mu\nu} \end{aligned}$$
or, written out
$$\displaystyle \begin{aligned} \begin{aligned} \Lambda^{\alpha\beta} &= h^{\alpha\nu}_{,\mu} h^{\beta\mu}_{,\nu} - h^{\mu\nu} h^{\alpha\beta}_{,\mu\nu} + {1 \over 2} g^{\alpha\beta} g_{\lambda\mu} h^{\lambda\nu}_{,\rho} h^{\rho\mu}_{,\nu} \\ &- (g^{\alpha\lambda} g_{\mu\nu}h^{\beta\nu}_{,\rho} h^{\mu\rho}_{,\lambda} + g^{\beta\lambda}g_{\mu\nu} h^{\alpha\nu}_{, \rho} h^{\mu\rho}_{,\lambda}) + g_{\lambda\mu}g^{\nu\rho} h^{\alpha\lambda}_{,\nu} h^{\beta\mu}_{,\rho} \\ & + {1 \over 8} (2 g^{\alpha\lambda} g^{\beta\mu} - g^{\alpha\beta} g^{\lambda\mu})(2 g_{\nu\rho} g_{\sigma\tau} - g_{\rho\sigma} g_{\nu\tau}) h^{\nu\tau}_{,\lambda} h^{\rho\sigma}_{,\mu} \, . \end{aligned} \end{aligned} $$
(5.8.32)
We see that the gravitational source terms contains products of the metric tensor that are at least quadratic in h and first and second derivatives. We write in obvious notation
$$\displaystyle \begin{aligned} \Lambda^{\alpha\beta} = \Lambda_2^{\alpha\beta}[h,h] + \Lambda_3^{\alpha\beta}[h,h,h] + \mathcal{O}(h^4) \end{aligned} $$
(5.8.33)
with
$$\displaystyle \begin{aligned} \begin{aligned} \Lambda_2^{\alpha\beta} =& - h^{\rho\sigma} \partial_{\rho\sigma} h^{\alpha\beta} + \frac{1}{2} \partial^\alpha h_{\rho\sigma} \partial^\beta h^{\rho\sigma} - \frac{1}{4} \partial^\alpha h \partial^\beta h + \partial_\sigma h^{\alpha\rho}(\partial^\sigma h^\beta_\rho + \partial_\rho h^{\beta\sigma}) \\ & - 2 \partial^{(\alpha} h_{\rho\sigma} \partial^\rho h^{\beta)\sigma} + \eta^{\alpha\beta} \left[ - \frac{1}{4} \partial_\tau h_{\rho\sigma} \partial^\tau h^{\rho\sigma} + \frac 1 8 \partial_\rho h \partial^\rho h + \frac{1}{2} \partial_\rho h_{\sigma\tau} \partial^\sigma h^{\rho\tau} \right] \, . \end{aligned} \end{aligned} $$
(5.8.34)
All indices are lowered and raised with the Minkowski metric η μν; h ≡ η αβ h αβ; the parenthesis around indices indicate symmetrization. Explicit expressions for $$\Lambda _3^{\alpha \beta }$$ and $$\Lambda _4^{\alpha \beta }$$ can be found in Blanchet and Faye (2001a).
Under certain assumptions the field equations (5.8.28) can formally be solved. Usually one imposes some ‘no incoming radiation’ condition of the form
$$\displaystyle \begin{aligned} {\partial \over \partial t} [h^{\alpha\beta}(t,\mathbf{x})] = 0 \qquad \mathrm{for} \ t \le -T_0 \, . \end{aligned} $$
(5.8.35)
Under this condition equation (5.8.28) is formally solved by
../images/447007_1_En_5_Chapter/447007_1_En_5_Equ127_HTML.png
(5.8.36)
with
../images/447007_1_En_5_Chapter/447007_1_En_5_Equ128_HTML.png
(5.8.37)
where the retarded time t R is given by
$$\displaystyle \begin{aligned} t_{\mathrm{R}} \equiv t - {\vert \mathbf{x} - \mathbf{x}' \vert \over c} \, . \end{aligned} $$
(5.8.38)