Test Bodies and Signals in a Riemann Space–Time

Abstract

In the previous chapters we have introduced a possible generalization of the space–time geometry based on the model of Riemann manifold.

Appendices

Exercises Chap. 5

5.1

Time-Dependent Spectral Shift A photon propagates along the (light-like) geodesics associated to the following line-element:

$$\begin{aligned} ds^2= c^2 dt^2- a^2(t) \left| d \varvec{x} \right| ^2, \end{aligned}$$

(5.43)

which describes a space–time with a time-dependent geometry. The photon is emitted at a time $t_e$ and received at a time $t_r$. Compute the frequency shift of the received photon with respect to the emitted one in terms of the geometric parameter a(t).

5.2

Hyperbolic Geodesic Motion Determine the time-like geodesic trajectories associated to a motion along the x axes, in the space–time manifold parametrized by the coordinates $x^0=ct$, $x^i=(x,y,z)$, and described by the following line-element:

$$\begin{aligned} ds^2= \left( t_0\over t\right) ^2 \left( c^2 dt^2 - dx^2\right) -dy^2 -dz^2, \end{aligned}$$

(5.44)

where $t_0$ is a constant.

Solutions

5.1

Solution In order to evaluate the frequency shift experienced by the photon along its geodesic path we shall use the condition of parallel transport of the four-momentum vector, given by Eq. (5.9).

We recall, first of all, that in the Minkowski space–time a photon of frequency $\omega$ is characterized by an energy $\mathcal{E}= \hbar \omega$ and a momentum $p^i= (\hbar \omega /c) n^i$, where $n^i$ is the unit vector pointing to the propagation direction. In the space–time described by the line-element (5.43) the four-momentum $p^\mu$ of the photon has then the following components:

$$\begin{aligned} p^0= {\hbar \omega \over c}, ~~~~~~~~~~~~~~~ p^i= {n^i\over a(t)} {\hbar \omega \over c} . \end{aligned}$$

(5.45)

The factor $a^{-1}$ appearing in the spatial part of the vector is prescribed by the minimal coupling principle, in order to satisfy the covariant version of the null normalization condition:

$$\begin{aligned} g_{\mu \nu } p^\mu p^\nu =\left( p^0\right) ^2 - a^2(t) \left| \varvec{p}\right| ^2=0. \end{aligned}$$

(5.46)

The non-vanishing components of the connection associated to the metric (5.43) are given by:

$$\begin{aligned} \varGamma _{0i}\,^j= {1\over ac} {d a \over dt} \delta _i^j, ~~~~~~~~~~ \varGamma _{ij}\,^0={a\over c} {d a \over dt} \delta _{ij} \end{aligned}$$

(5.47)

(we have used the definition (3.90)). By applying the geodesic equation (5.9) we then obtain

$$\begin{aligned} dp^0= d\left( \hbar \omega \over c\right)&= - \varGamma _{ij}\,^0 dx^i p^j \nonumber \\&=-{\hbar \omega \over c^2} {da\over dt} \delta _{ij} dx^in^j. \end{aligned}$$

(5.48)

Let us now recall that a light-like geodesic is characterized by a null space–time interval, $dx_\mu dx^\mu = ds^2=0$. A photon propagating along the spatial direction $n^i$, across a space–time geometry specified by Eq. (5.43), must then follow a trajectory which satisfies the differential condition

$$\begin{aligned} c dt \,n^i= a \,dx^i. \end{aligned}$$

(5.49)

Inserting this result into Eq. (5.48), using $\delta _{ij}n^in^j=1$, and dividing by $\hbar /c$, we obtain:

$$\begin{aligned} {d\omega \over \omega }= - {da\over a}. \end{aligned}$$

(5.50)

The integration of this equation immediately gives the time dependence of $\omega$ as a function of the time dependence of the geometric parameter a(t):

$$\begin{aligned} \omega (t)= {\omega _0\over a(t)}, \end{aligned}$$

(5.51)

where $\omega _0$ is an integration constant, representing the corresponding frequency in the Minkowski space–time (where $a=1$). The spectral shift between the emitted frequency $\omega _e\equiv \omega (t_e)$ and the received frequency $\omega _r\equiv \omega (t_r)$ is then fixed by the ratio:

$$\begin{aligned} {\omega _r\over \omega _e}={a(t_e)\over a(t_r)}. \end{aligned}$$

(5.52)

It may be noted, finally, that if $a(t_r)>a(t_e)$ then we obtain $\omega _r<\omega _e$, namely the received frequency is red-shifted with respect to the emitted one. This is a typical effect of the cosmological gravitational field which permeates our Universe on very large scales of distance, and which can be indeed described (in first approximation) by a geometry of the type (5.43) (see e.g. the books [12, 19, 24] quoted in the bibliography).

5.2

Solution The geometry of the manifold (5.44) is described by the metric

$$\begin{aligned}&g_{00}=\left( t_0\over t\right) ^2 = - g_{11}, ~~~~~~~~~~~ g_{22}=g_{33}=-1, \nonumber \\&g^{00}=\left( t\over t_0\right) ^2 = - g^{11}, ~~~~~~~~~~~ g^{22}=g^{33}=-1, \end{aligned}$$

(5.53)

and the non-zero components of the associated connection (defined by Eq. (3.90)) are given by:

$$\begin{aligned} \varGamma _{00}\,^0=\varGamma _{11}\,^0=\varGamma _{01}\,^1=\varGamma _{10}\,^1=-{1\over ct}. \end{aligned}$$

(5.54)

Let us explicitly write down the components of the geodesic equation (5.7), by setting $x^0=ct$ and recalling that the dot denotes differentiation with respect to the parameter $\tau$, which we can identify with the proper time along the particle trajectory:

$$\begin{aligned}&c \ddot{t}-{1\over ct} \left( c^2\dot{t}^2 + \dot{x}^2\right) =0, \end{aligned}$$

(5.55)

$$\begin{aligned}&\ddot{x}-{2\over t} \dot{x} \dot{t}=0, \end{aligned}$$

(5.56)

$$\begin{aligned}&\ddot{y}=0, \end{aligned}$$

(5.57)

$$\begin{aligned}&\ddot{z}=0. \end{aligned}$$

(5.58)

We are considering, in particular, a one-dimensional motion along the x axes. Equation (5.56) can thus be easily integrated, and gives

$$\begin{aligned} \dot{x}={\alpha \over t_0} t^2, \end{aligned}$$

(5.59)

where $\alpha$ is an integration constant with the dimensions of an acceleration (the parameter $t_0$ has been inserted for further convenience). Instead of integrating also Eq. (5.55) we now observe that a time-like geodesic must satisfy the normalization condition

$$\begin{aligned} g_{\mu \nu } \dot{x}^\mu \dot{x}^\nu = \left( t_0\over t\right) ^2 \left( c^2 \dot{t}^2 - \dot{x}^2 \right) = c^2, \end{aligned}$$

(5.60)

which, combined with Eq. (5.59), gives

$$\begin{aligned} c^2 \dot{t}^2 ={c^2\over t_0^2} t^2 +{\alpha ^2 \over t_0^2}t^4. \end{aligned}$$

(5.61)

Combining this result with Eq. (5.59) we can eliminate the proper-time variable $\tau$ (in terms of t), to obtain

$$\begin{aligned} {dx\over dt}= {\dot{x} \over \dot{t}}= {\alpha t \over \sqrt{1+{\alpha ^2 t^2\over c^2}}}. \end{aligned}$$

(5.62)

Another simple integration finally gives us the equation of the trajectory,

$$\begin{aligned} x(t)= x_0 +\int dt {\alpha t \over \sqrt{1+{\alpha ^2 t^2\over c^2}}}= x_0 +{c^2\over \alpha } \sqrt{1+{\alpha ^2 t^2\over c^2}}, \end{aligned}$$

(5.63)

where $x_0$ is an integration constant depending on the initial conditions. Note that this solution also automatically satisfies Eq. (5.55), as can be explicitly checked by differentiating with respect to $\tau$.

The above trajectory can be geometrically interpreted by squaring $x -x_0$, and moving ct to the left-hand side:

$$\begin{aligned} (x-x_0)^2 -c^2t^2={c^4\over \alpha ^2}. \end{aligned}$$

(5.64)

In the plane (x, ct) this is an hyperbola with center at the point of coordinates $x=x_0$ and $t=0$, and asymptotes given by the light-cone $x= x_0 \pm ct$. The time-like geodesics of the given geometry thus exactly reproduce the trajectories of a uniformly accelerated motion in Minkowski space, characterized by a four-acceleration vector with constant modulus $a_\mu a^\mu = \alpha ^2$.