### Summary

### 5.1 Invariance of Physical Laws

- Relativity is the study of how observers in different reference frames measure the same event.
- Modern relativity is divided into two parts. Special relativity deals with observers in uniform (unaccelerated) motion, whereas general relativity includes accelerated relative motion and gravity. Modern relativity is consistent with all empirical evidence thus far and, in the limit of low velocity and weak gravitation, gives close agreement with the predictions of classical (Galilean) relativity.
- An inertial frame of reference is a reference frame in which a body at rest remains at rest and a body in motion moves at a constant speed in a straight line unless acted upon by an outside force.
- Modern relativity is based on Einstein’s two postulates. The first postulate of special relativity is that the laws of physics are the same in all inertial frames of reference. The second postulate of special relativity is that the speed of light
*c*is the same in all inertial frames of reference, independent of the relative motion of the observer and the light source. - The Michelson-Morley experiment demonstrated that the speed of light in a vacuum is independent of the motion of Earth about the sun.

### 5.2 Relativity of Simultaneity

- Two events are defined to be simultaneous if an observer measures them as occurring at the same time (such as by receiving light from the events).
- Two events at locations a distance apart that are simultaneous for an observer at rest in one frame of reference are not necessarily simultaneous for an observer at rest in a different frame of reference.

### 5.3 Time Dilation

- Two events are defined to be simultaneous if an observer measures them as occurring at the same time. They are not necessarily simultaneous to all observers—simultaneity is not absolute.
- Time dilation is the lengthening of the time interval between two events when seen in a moving inertial frame rather than the rest frame of the events (in which the events occur at the same location).
- Observers moving at a relative velocity
*v*do not measure the same elapsed time between two events. Proper time $\text{\Delta}\tau $ is the time measured in the reference frame where the start and end of the time interval occur at the same location. The time interval $\text{\Delta}t$ measured by an observer who sees the frame of events moving at speed*v*is related to the proper time interval $\text{\Delta}\tau $ of the events by the equation:

$$\text{\Delta}t=\frac{\text{\Delta}\tau}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}=\gamma \text{\Delta}\tau ,$$

where

$$\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}.$$ - The premise of the twin paradox is faulty because the traveling twin is accelerating. The journey is not symmetrical for the two twins.
- Time dilation is usually negligible at low relative velocities, but it does occur, and it has been verified by experiment.
- The proper time is the shortest measure of any time interval. Any observer who is moving relative to the system being observed measures a time interval longer than the proper time.

### 5.4 Length Contraction

- All observers agree upon relative speed.
- Distance depends on an observer’s motion. Proper length ${L}_{0}$ is the distance between two points measured by an observer who is at rest relative to both of the points.
- Length contraction is the decrease in observed length of an object from its proper length ${L}_{0}$ to length
*L*when its length is observed in a reference frame where it is traveling at speed*v*. - The proper length is the longest measurement of any length interval. Any observer who is moving relative to the system being observed measures a length shorter than the proper length.

### 5.5 The Lorentz Transformation

- The Galilean transformation equations describe how, in classical nonrelativistic mechanics, the position, velocity, and accelerations measured in one frame appear in another. Lengths remain unchanged and a single universal time scale is assumed to apply to all inertial frames.
- Newton’s laws of mechanics obey the principle of having the same form in all inertial frames under a Galilean transformation, given by

$$x=x\prime +vt,\phantom{\rule{0.5em}{0ex}}y=y\prime ,\phantom{\rule{0.5em}{0ex}}z=z\prime ,\phantom{\rule{0.5em}{0ex}}t=t\prime .$$

The concept that times and distances are the same in all inertial frames in the Galilean transformation, however, is inconsistent with the postulates of special relativity. - The relativistically correct Lorentz transformation equations are

$$\begin{array}{ccc}\text{Lorentz transformation}\hfill & & \text{Inverse Lorentz transformation}\hfill \\ t=\frac{t\prime +vx\prime \text{/}{c}^{2}}{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}\hfill & & t\prime =\frac{t-vx\text{/}{c}^{2}}{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}\hfill \\ x=\frac{x\prime +vt\prime}{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}\hfill & & x\prime =\frac{x-vt}{\sqrt{1-{v}^{2}\text{/}{c}^{2}}}\hfill \\ y=y\prime \hfill & & y\prime =y\hfill \\ z=z\prime \hfill & & z\prime =z\hfill \end{array}$$

We can obtain these equations by requiring an expanding spherical light signal to have the same shape and speed of growth,*c*, in both reference frames. - Relativistic phenomena can be explained in terms of the geometrical properties of four-dimensional space-time, in which Lorentz transformations correspond to rotations of axes.
- The Lorentz transformation corresponds to a space-time axis rotation, similar in some ways to a rotation of space axes, but in which the invariant spatial separation is given by $\text{\Delta}s$ rather than distances $\text{\Delta}r,$ and that the Lorentz transformation involving the time axis does not preserve perpendicularity of axes or the scales along the axes.
- The analysis of relativistic phenomena in terms of space-time diagrams supports the conclusion that these phenomena result from properties of space and time itself, rather than from the laws of electromagnetism.

### 5.6 Relativistic Velocity Transformation

- With classical velocity addition, velocities add like regular numbers in one-dimensional motion: $u=v+u\prime ,$ where
*v*is the velocity between two observers,*u*is the velocity of an object relative to one observer, and $u\prime $ is the velocity relative to the other observer. - Velocities cannot add to be greater than the speed of light.
- Relativistic velocity addition describes the velocities of an object moving at a relativistic velocity.

### 5.7 Doppler Effect for Light

- An observer of electromagnetic radiation sees relativistic Doppler effects if the source of the radiation is moving relative to the observer. The wavelength of the radiation is longer (called a red shift) than that emitted by the source when the source moves away from the observer and shorter (called a blue shift) when the source moves toward the observer. The shifted wavelength is described by the equation:

$${\lambda}_{\text{obs}}={\lambda}_{s}\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}.$$

where ${\lambda}_{\text{obs}}$ is the observed wavelength, ${\lambda}_{s}$ is the source wavelength, and*v*is the relative velocity of the source to the observer.

### 5.8 Relativistic Momentum

- The law of conservation of momentum is valid for relativistic momentum whenever the net external force is zero. The relativistic momentum is $p=\gamma mu,$ where
*m*is the rest mass of the object,*u*is its velocity relative to an observer, and the relativistic factor is $\gamma =\frac{1}{\sqrt{1-\frac{{u}^{2}}{{c}^{2}}}}.$ - At low velocities, relativistic momentum is equivalent to classical momentum.
- Relativistic momentum approaches infinity as
*u*approaches*c*. This implies that an object with mass cannot reach the speed of light.

### 5.9 Relativistic Energy

- The relativistic work-energy theorem is ${W}_{\text{net}}=E-{E}_{0}=\gamma m{c}^{2}-m{c}^{2}=(\gamma -1)m{c}^{2}.$
- Relativistically, ${W}_{\text{net}}={K}_{\text{rel}}$ where ${K}_{\text{rel}}$ is the relativistic kinetic energy.
- An object of
*mass**m*at velocity*u*has kinetic energy ${K}_{\text{rel}}=(\gamma -1)m{c}^{2},$ where $\gamma =\frac{1}{\sqrt{1-\frac{{u}^{2}}{{c}^{2}}}}.$ - At low velocities, relativistic kinetic energy reduces to classical kinetic energy.
- No object with mass can attain the speed of light, because an infinite amount of work and an infinite amount of energy input is required to accelerate a mass to the speed of light.
- Relativistic energy is conserved as long as we define it to include the possibility of mass changing to energy.
- The total energy of a particle with mass
*m*traveling at speed*u*is defined as $E=\gamma m{c}^{2},$ where $\gamma =\frac{1}{\sqrt{1-\frac{{u}^{2}}{{c}^{2}}}}$ and*u*denotes the velocity of the particle. - The rest energy of an object of mass
*m*is ${E}_{0}=m{c}^{2},$ meaning that mass is a form of energy. If energy is stored in an object, its mass increases. Mass can be destroyed to release energy. - We do not ordinarily notice the increase or decrease in mass of an object because the change in mass is so small for a large increase in energy. The equation ${E}^{2}={(pc)}^{2}+{(m{c}^{2})}^{2}$ relates the relativistic total energy
*E*and the relativistic momentum*p*. At extremely high velocities, the rest energy $m{c}^{2}$ becomes negligible, and $E=pc.$