### 28.1 Einstein’s Postulates

- Relativity is the study of how different observers measure the same event.
- Modern relativity is divided into two parts. Special relativity deals with observers who are in uniform (unaccelerated) motion, whereas general relativity includes accelerated relative motion and gravity. Modern relativity is correct in all circumstances and, in the limit of low velocity and weak gravitation, gives the same predictions as classical 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 on by an outside force.
- Modern relativity is based on Einstein’s two postulates. The first postulate of special relativity is the idea that the laws of physics are the same and can be stated in their simplest form in all inertial frames of reference. The second postulate of special relativity is the idea that the speed of light $c$ is a constant, independent of the relative motion of the source.
- The Michelson-Morley experiment demonstrated that the speed of light in a vacuum is independent of the motion of the Earth about the Sun.

### 28.2 Simultaneity And 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 phenomenon of time passing slower for an observer who is moving relative to another observer.
- Observers moving at a relative velocity $v$ do not measure the same elapsed time for an event. Proper time $\mathrm{\Delta}{t}_{0}$ is the time measured by an observer at rest relative to the event being observed. Proper time is related to the time $\mathrm{\Delta}t$ measured by an Earth-bound observer by the equation
$$\mathrm{\Delta}t=\frac{{\mathrm{\Delta}t}_{0}}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}={\gamma \mathrm{\Delta}t}_{0},$$
where

$$\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}.$$ - The equation relating proper time and time measured by an Earth-bound observer implies that relative velocity cannot exceed the speed of light.
- The twin paradox asks why a twin traveling at a relativistic speed away and then back towards the Earth ages less than the Earth-bound twin. The premise to the paradox is faulty because the traveling twin is accelerating. Special relativity does not apply to accelerating frames of reference.
- Time dilation is usually negligible at low relative velocities, but it does occur, and it has been verified by experiment.

### 28.3 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. Earth-bound observers measure proper length when measuring the distance between two points that are stationary relative to the Earth.
- Length contraction $L$ is the shortening of the measured length of an object moving relative to the observer’s frame:$${\text{L=L}}_{0}\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}=\frac{{L}_{0}}{\gamma}.$$

### 28.4 Relativistic Addition of Velocities

- 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 speed:
$$\text{u=}\frac{v+u\prime}{1+\frac{v\text{u}\prime}{{c}^{2}}}$$
- 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
$${\lambda}_{\text{obs}}{\text{=\lambda}}_{s}\sqrt{\frac{1+\frac{u}{c}}{1-\frac{u}{c}}}$$${\lambda}_{\text{obs}}$ is the observed wavelength, ${\lambda}_{s}$ is the source wavelength, and $u$ is the relative velocity of the source to the observer.

### 28.5 Relativistic Momentum

- The law of conservation of momentum is valid whenever the net external force is zero and for relativistic momentum. Relativistic momentum $p$ is classical momentum multiplied by the relativistic factor $\gamma $.
- $p=\text{\gamma mu}$, where $m$ is the rest mass of the object, $u$ is its velocity relative to an observer, and the relativistic factor $\gamma =\frac{1}{\sqrt{1-\frac{{u}^{2}}{{c}^{2}}}}$.
- 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.
- Relativistic momentum is conserved, just as classical momentum is conserved.

### 28.6 Relativistic Energy

- Relativistic energy is conserved as long as we define it to include the possibility of mass changing to energy.
- Total Energy is defined as: $E={\mathrm{\gamma mc}}^{2}$, where $\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}$.
- Rest energy is ${E}_{0}={\mathrm{mc}}^{2}$, 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 relativistic work-energy theorem is ${W}_{\text{net}}=E-{E}_{0}=\gamma {\mathrm{mc}}^{2}-{\mathrm{mc}}^{2}=\left(\gamma -1\right){\mathrm{mc}}^{2}$.
- Relativistically, ${W}_{\text{net}}={\text{KE}}_{\text{rel}}$ , where ${\text{KE}}_{\text{rel}}$ is the relativistic kinetic energy.
- Relativistic kinetic energy is ${\text{KE}}_{\text{rel}}=\left(\gamma -1\right){\mathrm{mc}}^{2}$, where $\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}$. 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.- The equation ${E}^{2}=(\mathrm{pc}{)}^{2}+({\mathrm{mc}}^{2}{)}^{2}$ relates the relativistic total energy $E$ and the relativistic momentum $p$. At extremely high velocities, the rest energy ${\mathrm{mc}}^{2}$ becomes negligible, and $E=\mathrm{pc}$.