Accelerated Reference Frames

Abstract

This chapter begins with an introduction to the formalism used to project four-dimensional spacetime into a 3-dimensional spatial 3-space. Then we apply this formalism to deduce the spatial geometry in a rotating reference frame and discuss Ehrenfest’s paradox. Also we show that it is impossible to Einstein synchronize clocks around a closed path in a rotating frame because this leads to a contradiction in a non-rotating frame. Gravitational time dilation and frequency shift, and also the Sagnac experiment are discussed. Finally we give an introduction to special relativistic kinematic in a uniformly accelerated reference frame in flat spacetime. It is pointed out that an observer experiences an acceleration of gravity in such a frame.

Exercises

• 4.1. Relativistic rotating disc

A disc rotates with constant angular velocity in $\omega$ in its own plane and around a fixed axis $A$. The axis is chosen to be the origin in a non-rotating Cartesian coordinate system $(x,y)$ with coordinate clocks showing $t$. (The $z$-coordinate is kept constant from now on). The motion of a given point on the disc can be expressed as

$$x = r\cos (\omega t + \phi ),y = r\sin (\omega t + \phi ),$$

(4.115)

where $(r,\phi )$ are coordinates specifying the point at the disc.

• (a) An observer is able to move on the disc and performs measurements of distance between neighbouring points at different locations on the disc. The measurements are performed when the observer is stationary with respect to the disc. The result is assumed to be the same as that measured in an inertial frame with the same velocity as the observer at the time of the measurement. The lengths measured by the observer are given by

$${\text{d}}\ell^{2} = f_{1} (r,\phi ){\text{d}}r^{2} + f_{2} (r,\phi ){\text{d}}\phi^{2} .$$

(4.116)

Find $f_{1} (r,\phi )$ and $f_{2} (r,\phi )$.

We now assume that the observer measures the distance from the axis $A$ to a point $(R,0)$ along the line $\phi = 0$, by adding the result of measurements between neighbouring points. What is the result the observer finds? Furthermore the observer measures the distance around the circle $r = R$. What is then found? In what way, based on this result, is it possible to deduce that the metric considered by the observer is non-Euclidean? Will the observer find a negative or positive curvature of the disc?

We introduce coordinates $(\tilde{t},\tilde{x},\tilde{y})$ that follow the rotating disc. They are given by

$$\tilde{t} = t,\,\tilde{x} = r\cos \phi ,\,\tilde{y} = r\sin \phi .$$

(4.117)

• (b) Find the invariant interval ${\text{d}}s^{2} = {\text{d}}x^{2} + {\text{d}}y^{2} - c^{2} {\text{d}}t^{2}$ in terms of the coordinates $(\tilde{t},\tilde{x},\tilde{y})$.

• (c) Light signals are sent from the axis $A$. How will the paths of the light signals be as seen from the $(\tilde{x},\tilde{y})$ system? Draw a figure that illustrates this. A light signal with the frequency $\nu_{0}$ is received by the observer in $r = R$, $\phi = 0$. Which frequency $\nu$ will be measured by the observer?

• (d) We now assume that standard clocks measuring proper time are tightly packed around the circle $r = R$. The clocks are at rest on the disc. We now want to synchronize the clocks and start out with a clock at the point $(R,0)$. The clocks are then synchronized in the direction of increasing $\phi$ in the following way: When a clock is tuned at a point $\phi$, the clock at the neighbouring point $\phi + d\phi$ is also tuned so that it shows the same time at simultaneity in the instantaneous rest frame of the two clocks.

Show that there is a problem with synchronization when this process is performed around the entire circle, by the fact that the clock we started out with is no longer synchronous with the neighbouring clock which is tuned according to the synchronization process. Find the time difference between these two clocks.

• (e) Locally around a point $(r,\phi ,t)$ we can define an inertial system being an instantaneous rest frame of the point $(r,\phi )$ on the disc. We introduce an orthonormal set of basis vectors $\vec{e}_{{\hat{\lambda }}}$, $\vec{e}_{{\hat{\eta }}}$ and $\vec{e}_{{\hat{\xi }}}$ in this frame. The vector $\vec{e}_{{\hat{\lambda }}}$ points along the time axis of the system, $\vec{e}_{{\hat{\xi }}}$ points radially and $e_{{\hat{\eta }}}$ tangentially. Find the vectors expressed by $\vec{e}_{t}$, $\vec{e}_{x}$ and $\vec{e}_{y}$.

• 4.2. Uniformly accelerated system of reference

We will now study a coordinate system $(t,x)$ co-moving with a uniformly accelerated reference frame, AF, in a 2-dimensional Minkowski space. The connection with a Cartesian coordinate system $(T,X)$ co-moving with an instantaneous inertial rest frame, IF, of AF at the point of time $T = 0$ is given by the coordinate transformation

$$T = x\sinh (at),X = x\cosh (at)$$

(4.118)

where $a$ is a constant.

• (a) Draw the coordinate lines $t =$ constant and $x =$ constant in a $(T,X)$-diagram.

• (b) Find the line-element ${\text{d}}s^{2} = - {\text{d}}T^{2} + {\text{d}}X^{2}$ expressed by $t$ and $x$.

• (c) We now assume that a particle has a path in spacetime so that it follows one of the curves $x =$ constant. Such a motion is called hyperbolic motion. Why?

  Show that the particle has constant acceleration when the acceleration is measured in the instantaneous rest frame of the particle. Find the acceleration of the rod has constant acceleration and the rod $g$. Find also the velocity and acceleration of the particle in the system $(T,X)$.

• (d) Show that at any point on the trajectory of a reference particle in AF the direction of the coordinate axes in the $(T,X)$-system will overlap with the time and spatial axis of the instantaneous rest frame of the particle. Explain why it is possible to see from the line element that the $X$-coordinate measures length along the spatial axis, whereas the $T$-coordinate, which is the coordinate time, is in general not the proper time of the particle. For what value of $X$ is the coordinate time equal to the proper time?

  The $(t,x)$-coordinate system can be considered as an attempt to construct, from the instantaneous rest frames along the path, a coordinate system covering the entire spacetime. Explain why this is not possible for the entire space. (Hint: There is a coordinate singularity at a certain distance from the trajectory of the particle).

• (e) A rod is moving in the direction of its own length. At the time $T = 0$ the rod is at rest, but still accelerated. The length of the rod measured in the stationary system is $L$ at this time. The rod moves so that the forwards point of the rod has constant rest acceleration measured in the instantaneous rest frame.

We assume that the acceleration of the rod finds place so that the infinitesimal distance ${\text{d}}\ell$ between neighbouring points on the rod measured in the instantaneous rest frame is constant. Find the motion of the rear point of the rod in the stationary reference system. Why is there a maximal length of the rod, $L_{ \hbox{max} }$ ?

If the rear point of the rod has constant acceleration and the rod is accelerated as previously in this exercise, then is there a maximal value of $L$ ?

• 4.3. Uniformly accelerated space ship

  • (a) A spaceship leaves the Earth at the time $T = 0$ and moves with a constant acceleration $g$, equal to the gravitational constant at the Earth, into space. Find how far the ship has travelled during 10 years of proper time of the ship.

  • (b) Radio signals are sent from the Earth towards the spaceship. Show that signals that are sent after a given time $T$ will never reach the ship (even if the signals travel with the speed of light). Find $T$. At what time are the signals sent from the Earth if they reach the ship after 10 years (proper time of the ship)?

  • (c) Calculate the frequency of the radio signals received by the ship, given by the frequency $\nu_{0}$ (emitter frequency) and the time $t_{0}$ (emitter time). Investigate the behaviour of the frequency when the proper time on the space ship $\tau \to \infty$.

• 4.4. Light emitted from a point source in a gravitational field

A point-like light source is at the position $x = x_{1} ,y = 0$ in a uniformly accelerated reference frame AF. A photon is emitted from the source at a point of time $t = 0$. It is emitted in the $\left( {x,y} \right) -$ plane in a direction making an angle $\theta_{0}$ with the x-axis.

Find the equation for the path followed by the photon, and identify the nature of the trajectory.

• 4.5. Geometrical optics in a gravitational field

It was shown in Eq. (4.99) that light does not move along a straight path in the gravitational field experienced in a uniformly accelerated reference frame AF, but along a circular trajectory.

• (a) A sphere seen from above. The observer P is at a height $x = x_{1}$. The centre of the sphere is at a distance b vertically beneath P. Light is emitted from the surface of the body. It moves along a circular path with radius $R = x_{1} /\sin \theta$, where $\theta$ is the angle between the light path and the x-axis at the position of the observer. The corresponding angle without a gravitational field would be $\theta_{0}$ given by $\sin \theta_{0} = r/b$, where r is the radius of the sphere. The acceleration of gravity at the position of the observer is $g/c^{2} = 1/x_{1}$.

  Calculate how the angle $\theta$ depends upon g, r and b.

• (b) A sphere seen from below. Same as (a) but with the observer below the sphere.

• (c) An experiment. Place a camera one metre above the centre of a sphere of radius 10 cm and another one a metre below in the gravitational field of the Earth. Calculate the difference of $\theta$ in photographs taken with the two cameras.