Technical warm-up

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Uniformly accelerated observer, Rindler metric

Einstein's equivalence principle states that locally a gravitational field cannot be distinguished from a non-inertial frame of reference. Therefore a number of effects of General Relativity, such as time dilation in a gravitational field and formation of horizons, can be studied in the frame of Special Theory of Relativity when considering uniformly accelerated observers.

Problem 1: a particle in homogeneous electric field

Derive the equation of motion $x(t)$ of a charged particle in Minkowski space in a uniform electric field without initial velocity. Show that its acceleration is constant.

Problem 2: (unexpected?) horizons!

What region of spacetime is unobservable for such an accelerated observer? In what region is this observer unobservable?

Problem 3: Rindler metric

Consider the set of particles, which move with constant accelerations $a=const>0$ in Minkowski space, with initial conditions at time $t=0$ set as $x=\rho=c^{2}/a$. Let $\tau$ be the proper time of these particles in the units of $\rho/c$. What region of spacetime is parametrized by the pair of positive numbers $(\tau,\rho)$? Express the metric in this region in the coordinates $(\rho,\varphi)$, where $\varphi=c\tau/\rho$. This is the Rindler metric.

For more details see textbooks, e.g.

Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald, Gravitation, San Francisco: W. H. Freeman, 1973, ISBN 978-0-7167-0344-0;

G. `t Hooft, Introduction to General Relativity, Caputcollege 1998, ISBN 978-1589490000;

Padmanabhan T. Gravitation: Foundations and Frontiers. CUP, 2010, ISBN 9780521882231.

Metric in curved spacetime

We see here, how, given an arbitrary metric tensor, to determine physical distance between points, local time and physical velocity of a particle in an arbitrary frame of reference.

This problem, though fundamentally important, is necessary in full form only for consideration of particle dynamics in the Kerr metric. In order to analyze the dynamics in the Schwarzschild metric, it suffices to answer all the questions with a substantially simplifying condition $g_{0\alpha}=0$, where $\alpha=1,2,3$ (see the last of the problems).

Let the spacetime metric have the general form \[ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}.\] Coordinates are arbitrary and do not carry direct metrical meaning. An observer, stationary in a given coordinate frame, has 4-velocity $u^{\mu}=(u^{0},0,0,0)$, and the interval determines his proper "local" time \[c^2 d\tau^{2}=ds^{2}=g_{00}(dx^{0})^2.\] An observer in point A, with coordinates $x^\mu$, determines the physical "radar" distance to an infinitely close point $B$, with coordinates $x^{\mu}+dx^{\mu}$, in the following way. She sends a light beam to $B$ and measures the time it takes for the reflected beam to come back. Then distance to $B$ is half the proper time she waited from emission to detection times $c$. It is also natural for her to consider the event of the beam reflection in $B$ to be simultaneous with the middle of the infinitely small 4-distance between the events of emission and detection of light beam in $A$.

Measuring distances

Problem 4: measuring distances

Find the proper (physical) distance $dl$ between two events with coordinates $x^\mu$ and $x^\mu+dx^{\mu}$.

Problem 5: simultaneous events

Find the difference between coordinate times of two infinitely close simultaneous events.

Problem 6: observing motion

Let a particle's world line be $x^{\mu}(\lambda)$. What is the proper time interval $\delta\tau$ of a stationary observer, in which this particle covers distance from $x^{\mu}$ to $x^\mu+dx^\mu$?

Problem 7: proper velocity and Lorentz factor

Proper (or physical) velocity $v$ of the particle is defined as $dl/\delta\tau$. Express it through the $4$-velocity of the particle and through its coordinate velocity $dx^{\alpha}/dx^{0}$; find the interval along the world line $ds$ in terms of $v$ and local time $\delta\tau$.

Problem 8: simplification: static case

How are all the previous answers simplified if $g_{0\alpha}=0$?

A coherent exposition can be found in e.g.

L.D. Landau, E.M. Lifshitz (1975). The Classical Theory of Fields. Vol. 2 (4th ed.). Butterworth-Heinemann. ISBN 978-0-7506-2768-9