# Learn Relativity and Gravitation with Problem Book: Free 15 Solutions

## Problem Book in Relativity and Gravitation: A Unique Resource for Students and Teachers

Are you interested in learning about one of the most fascinating theories of modern physics? Do you want to challenge yourself with some of the most intriguing problems in the fields of special and general relativity, gravitation, relativistic astrophysics, and cosmology? Do you want to have access to a comprehensive collection of problems with detailed solutions that can help you master these topics? If you answered yes to any of these questions, then you should definitely check out Problem Book in Relativity and Gravitation, a classic text by four leading experts in the field: Alan P. Lightman, William H. Press, Richard H. Price, and Saul A. Teukolsky.

## Problem Book in Relativity and Gravitation free 15

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In this article, we will give you an overview of what this book is, who are the authors, why is it important and useful, and how to use it. We will also give you a glimpse of some of the problems and solutions that you can find in the book, covering topics such as kinematics, dynamics, coordinate transformations, tensors, curved space-time, black holes, gravitational waves, stellar structure, cosmological models, and cosmic microwave background radiation. Finally, we will tell you how to get a free copy of the book, so you can start solving the problems yourself.

## Special-Relativistic Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the causes of their motion. In special relativity, kinematics is concerned with how the measurements of time, length, velocity, and angle change when observed from different inertial frames of reference, which are frames that move at constant velocities relative to each other. Special relativity also introduces the concept of four-vectors, which are quantities that have four components: one for time and three for space. Four-vectors are useful because they are invariant under Lorentz transformations, which are the mathematical rules that relate the measurements of different inertial observers.

### Lorentz transformations and four-vectors

One of the most famous consequences of special relativity is that time and length are not absolute, but depend on the state of motion of the observer. This means that two events that happen at the same time and place for one observer may happen at different times and places for another observer moving at a different velocity. This also means that an object that has a certain length for one observer may appear to be shorter or longer for another observer moving at a different velocity. These effects are known as time dilation and length contraction, respectively.

To quantify these effects, we can use the Lorentz transformations, which are equations that relate the coordinates of events as measured by two inertial observers moving along a common axis with a relative velocity v. The Lorentz transformations are given by:

$$x' = \gamma (x - vt)$$

$$y' = y$$

$$z' = z$$

$$t' = \gamma (t - vx/c^2)$$

where x, y, z, t are the coordinates of an event as measured by one observer, x', y', z', t' are the coordinates of the same event as measured by another observer moving with velocity v along the x-axis, c is the speed of light in vacuum, and $\gamma$ is a factor given by:

$$\gamma = \frac1\sqrt1 - v^2/c^2$$

The factor $\gamma$ is always greater than or equal to one, and it approaches infinity as v approaches c. This means that as an object moves faster and faster, its length along the direction of motion contracts more and more, and its time dilates more and more.

A useful way to represent events in space-time is to use four-vectors, which are vectors that have four components: one for time and three for space. For example, the position four-vector of an event is given by:

$$x^\mu = (ct, x, y, z)$$

The advantage of using four-vectors is that they are invariant under Lorentz transformations, meaning that their magnitude does not change when observed from different inertial frames. The magnitude of a four-vector is defined as:

$$x^\mu x_\mu = (ct)^2 - x^2 - y^2 - z^2$$

This quantity is also called the invariant interval between two events, and it can be positive, negative, or zero depending on whether the events are time-like separated (meaning that they can be causally connected by a signal traveling slower than light), space-like separated (meaning that they cannot be causally connected by any signal), or light-like separated (meaning that they can be causally connected by a signal traveling at the speed of light).

An example of a problem involving Lorentz transformations and four-vectors is:

A spaceship travels at a constant speed of 0.8c from Earth to a star 4 light-years away. How long does the trip take according to the spaceship's clock? How long does it take according to the Earth's clock? What is the length of the spaceship as measured by an observer on Earth?

The solution is:

We can use the Lorentz transformations to relate the coordinates of the events of departure and arrival as measured by the spaceship and the Earth. Let us choose the origin of both frames to coincide at the event of departure, and let us denote by x and t the coordinates of the event of arrival as measured by the Earth, and by x' and t' the coordinates of the same event as measured by the spaceship. Then we have:

$$x = 4 \text light-years$$

$$x' = 0$$

$$t = ?$$

$$t' = ?$$

Using the Lorentz transformations, we can solve for t and t' as follows:

$$t' = \gamma (t - vx/c^2) = \gamma (t - 0.8ct/c^2) = \gamma (1 - 0.8) t = 0.6 \gamma t$$

$$x = \gamma (x' + vt') = \gamma (0 + 0.8ct') = 0.8 \gamma c t'$$

Substituting x = 4 light-years and solving for t', we get:

$$t' = \fracx0.8 \gamma c = \frac40.8 \gamma \text years$$

where $\gamma = 1 / \sqrt1 - v^2/c^2 = 1 / \sqrt1 - 0.8^2 = 5/3$. Therefore:

$$t' = \frac40.8 (5/3) \text years = 1.5 \text years$$

This is the time elapsed according to the spaceship's clock. To find the time elapsed according to the Earth's clock, we can use the relation:

$$t = \fract'0.6 \gamma = \frac1.50.6 (5/3) \text years = 2.5 \text years$$

This is longer than the time measured by the spaceship, as expected from time dilation.

To find the length of the spaceship as measured by an observer on Earth, we can use the formula for length contraction:

$$L = L_0 / \gamma$$

where L is the contracted length, L_0 is the proper length (the length measured in the rest frame of the spaceship), and $\gamma$ is the same factor as before. Assuming that the proper length of the spaceship is 100 m, we get:

$$L = 100 / (5/3) \text m = 60 \text m$$

This is shorter than the proper length, as expected from length contraction.

### Doppler effect and aberration

Another consequence of special relativity is that the frequency and wavelength of light change when observed from different inertial frames. This effect is called the relativistic Doppler effect, and it is analogous to the classical Doppler effect for sound waves, but with some important differences.

The relativistic Doppler effect occurs when a source of light and an observer are moving towards or away from each other with a relative velocity v. The frequency f and wavelength λ of light are related by c = fλ, where c is the speed of light in vacuum, which is constant for all inertial observers. Therefore, if f or λ change due to motion, the other must change as well to keep c constant.

The formula for the relativistic Doppler effect for motion along a common axis is given by:

$$f_r = f_s\sqrt\frac1 + v/c1 - v/c$$

where f_r is the frequency detected by the receiver (observer), f_s is the frequency emitted by the source, and v is the relative velocity of the source and receiver, positive if they are moving away and negative if they are moving towards each other. This formula can also be written in terms of wavelengths as:

$$\lambda_r = \lambda_s\sqrt\frac1 - v/c1 + v/c$$

where λ_r is the wavelength detected by the receiver and λ_s is the wavelength emitted by the source.

The relativistic Doppler effect causes a redshift (increase in wavelength and decrease in frequency) when the source and receiver are moving away from each other, and a blueshift (decrease in wavelength and increase in frequency) when they are moving towards each other. The amount of shift depends on the relative velocity v, and it becomes more pronounced as v approaches c.

A related effect is the aberration of light, which is the change in the apparent direction of a light source due to the relative motion of the source and the observer. The formula for the aberration angle θ, which is the angle between the actual direction of the source and the apparent direction as seen by the observer, is given by:

$$\tan \theta = \fracv \sin \phic + v \cos \phi$$

where v is the relative velocity of the source and observer, c is the speed of light in vacuum, and φ is the angle between the actual direction of the source and the direction of motion of the observer.

The aberration of light causes a light source to appear shifted towards the direction of motion of the observer, and away from the direction of motion of the source. The amount of shift depends on the relative velocity v and the angle φ, and it becomes more pronounced as v approaches c or φ approaches 90.

An example of a problem involving Doppler effect and aberration is:

A star emits light with a wavelength of 500 nm. An observer on Earth sees the star with a wavelength of 550 nm. The star is located at an angle of 30 from the direction of motion of Earth around the Sun, which is 30 km/s. What is the radial velocity of the star (the component of its velocity along the line of sight)? What is the aberration angle of the star?

The solution is:

We can use the formula for the relativistic Doppler effect to find the radial velocity v_r of the star:

$$\lambda_r = \lambda_s\sqrt\frac1 - v_r/c1 + v_r/c$$

Substituting $\lambda_r = 550$ nm, $\lambda_s = 500$ nm, and c = 3 10^8 m/s, we get:

$$550 = 500\sqrt\frac1 - v_r/(3 \times 10^8)1 + v_r/(3 \times 10^8)$$

Squaring both sides and simplifying, we get:

$$v_r = \frac3 \times 10^8 (0.1)1.21 \text m/s = 2.48 \times 10^7 \text m/s$$

This is positive, meaning that the star is moving away from Earth.

To find the aberration angle θ, we need to consider that Earth's velocity v_e also contributes to the apparent shift in direction of the star. The total velocity v_t of Earth relative to the star is given by:

$$v_t = v_e + v_r$$

where v_e = 30 km/s is Earth's velocity around the Sun, and v_r = 2.48 10^7 m/s is the radial velocity of the star. The angle φ between v_e and v_t is given by:

$$\cos \phi = \fracv_ev_t$$

Substituting v_e = 30 km/s and v_t = (v_e^2 + v_r^2)^1/2, we get:

$$\cos \phi = \frac30\sqrt30^2 + (2.48 \times 10^7)^2 \text km/s = 0.0012$$

Therefore:

$$\phi = \cos^-1 0.0012 = 89. 9$

Using the formula for the aberration angle θ, we get:

$$\tan \theta = \fracv_t \sin \phic + v_t \cos \phi = \frac(30 + 2.48 \times 10^7) \sin 89.9(3 \times 10^8) + (30 + 2.48 \times 10^7) \cos 89.9$$

Therefore:

$$\theta = \tan^-1 0.0826 = 4.7$$

This means that the star appears to be shifted by 4.7 towards the direction of motion of Earth.

### Relativistic optics

Relativistic optics is the branch of physics that deals with the interaction of light and matter at relativistic speeds, that is, speeds comparable to or greater than the speed of light in vacuum. Relativistic optics encompasses a variety of phenomena that arise from the effects of special relativity and quantum electrodynamics on the propagation, reflection, refraction, polarization, scattering, and emission of light.

Some examples of relativistic optical phenomena are:

Light deflection by gravitational fields: According to general relativity, gravity can bend space-time, and thus affect the path of light rays. This effect can cause gravitational lensing, which is the distortion and magnification of distant light sources by massive objects such as galaxies or black holes.

Gravitational redshift: According to general relativity, gravity can also affect the frequency and wavelength of light. This effect can cause gravitational redshift, which is the decrease in frequency and increase in wavelength of light as it escapes from a gravitational field.

Lorentz contraction of light beams: According to special relativity, objects moving at relativistic speeds undergo length contraction along their direction of motion. This effect can cause Lorentz contraction of light beams, which is the decrease in diameter and increase in intensity of light beams as they propagate at relativistic speeds.

Relativistic beaming: According to special relativity, objects moving at relativistic speeds also undergo time dilation and Doppler shift. This effect can cause relativistic beaming, which is the enhancement and collimation of light emission along the direction of motion of a relativistic source.

Vacuum birefringence: According to quantum electrodynamics, vacuum is not a perfectly empty space, but a fluctuating sea of virtual particles that can be affected by external fields. This effect can cause vacuum birefringence, which is the splitting and polarization of light as it propagates through a strong magnetic field in vacuum.

An example of a problem involving relativistic optics is:

A laser beam with a wavelength of 800 nm and a diameter of 10 cm is focused by a lens onto a thin foil target. The target is accelerated by the laser radiation pressure to a speed of 0.9c. What is the wavelength and diameter of the laser beam as seen by an observer at rest with respect to the target?

The solution is:

We can use the formulas for the relativistic Doppler effect and Lorentz contraction to find the wavelength and diameter of the laser beam as seen by the target. Let us denote by λ_0 and D_0 the wavelength and diameter of the laser beam in the rest frame of the source (the lens), and by λ_t and D_t the wavelength and diameter of the laser beam in the rest frame of the target. Then we have:

$$\lambda_t = \lambda_0\sqrt\frac1 + v/c1 - v/c$$

where v = 0.9c is the relative velocity between the source and target frames, and c = 3 10^8 m/s is the speed of light in vacuum. Substituting λ_0 = 800 nm and v = 0.9c, we get:

$$\lambda_t = 800 \times 10^-9\sqrt\frac1 + 0.91 - 0.9 \text m = 5.66 \times 10^-6 \text m$$

This is a redshift, meaning that the laser beam appears to have a longer wavelength and lower frequency in the target frame.

$$D_t = D_0 / \gamma$$

where $\gamma = 1 / \sqrt1 - v^2/c^2 = 1 / \sqrt1 - 0.9^2 = 2.29$ is the Lorentz factor. Substituting D_0 = 10 cm and $\gamma = 2.29$, we get:

$$D_t = 10 / 2.29 \text cm = 4.37 \text cm$$

This is a Lorentz contraction, meaning that the laser beam appears to have a smaller diameter and higher intensity in the target frame. 71b2f0854b