Relativity

## Big Picture

When first published by Albert Einstein, the theories of special and general relativity went against much of what physicists had believed for hundreds of years. However, both theories have made predictions that have been proven to be correct in many experiments performed over the past century.

In classical mechanics, all speeds are measured relative to another reference frame. However, the speed of light was proven to be the same when measured from any reference frame. In order to maintain a constant speed of light, special relativity describes how the passage of time and distance change depending on the observer. The effects of special relativity are only visible at speeds approaching the speed of light, so the classical view of motion is still a very good approximation of motion at everyday speeds. Special relativity also tells us that the speed of light is the maximum speed anything can travel.

Although Newton’s law of universal gravitation explained the effects of gravity, general relativity actually explains how gravity works in the universe. General relativity geometrically explains how matter and radiation are effected in a gravitational field.

## Key Terms

Special Relativity: A theory stating the speed of light and the laws of physics is the same for every reference frame.

General Relativity: A theory stating gravity is not a force but a curvature in spacetime.

Spacetime: A combination of the three spatial dimensions with time as a fourth dimension.

## Special Relativity

### Postulates of Special Relativity

The two pillars of the theory of special relativity are:

• Light always travels at the same speed $$(c = 3 × 10^8 m/s)$$ regardless of the relative motion of the source of the light to an observer. This means that if a car traveling at 20 m/s shines a beam of light, the speed of that beam is c, not c + 20 m/s.
• The fundamental laws that determine the state of a system will always be the same in any non-accelerating (inertial) reference frame. From inside the system, it is impossible to determine if a closed system is at rest or moving uniformly because in both cases everything will seem exactly the same. It is impossible to measure your speed without measuring it relative to another external system.

### Time Dilation and Length Contraction

When dealing with light, speed is constant, so the ratio of distance to time must remain constant to all observers ($$c = \frac{\text{distance}}{\text{time}}$$). Imagine that we have a simple clock that is just a flash of light bouncing between two mirrors. If we put one on the ground and one in spaceship that is going at relativistic speeds, we will find that the clock in the spaceship will appear to move slower than the one on the ground. With each cycle, the flash of light is traveling at the same speed in the spaceship and on the ground (speed of light is constant). However, the flash of light in the spaceship is traveling farther than the one on the ground. Since the ratio of distance and time must remain constant, if the distance traveled increases, the time it takes to travel must also increase.

Below is a diagram that will illustrate this concept. The light clock on the left is the stationary clock and the one on the right is moving with speed v. The flash of light in the moving clock must cover a greater distance ($$\sqrt{(ct)^2 + (vt^1)^2}$$ ) at the same speed (c) from the perspective of a stationary observer outside the spaceship. Therefore, for a stationary observer, the flash of light appears to take more time to travel between the mirrors.

Stationary clock

Clock moving with speed v

Image Credit: Svjo, CC-BY-SA 3.0

Similarly, when moving at speeds comparable to the speed of light, an object’s length will noticeably contract (shorten) in the direction of motion to an observer in another reference frame. This concept stems from time dilation.

### Relativistic Mass

In special relativity, the mass of an object change depending on how fast an object is moving relative an observer. The relativistic mass of an object includes both the rest and kinetic energy of an object.

Since the development of special relativity and the realization that mass is a form of energy, the conservation laws for mass and energy were combined into one conservation law, the conservation of mass-energy.

Relativity cont.

## Special Relativity (cont.)

### Relativistic Kinetic Energy

When Einstein developed his theory of special relativity, he also revised the equations determining the energy of a moving object. At low speeds, the equation for kinetic energy used in classical mechanics is a good approximation, but at relativistic speeds, the energy of the moving object is much greater. Based on his equation, it is impossible for an object to move faster than the speed of light because it would take an infinite amount of energy to accelerate an object to the speed of light. The famous equation for mass-energy equivalence can be derived by applying the equation for relativistic energy to a stationary object.

At first glance, the twins paradox is a famous thought experiment that seems to contradict special relativity. The situation is: if there are two twins starting on Earth, and one moves away on a space ship at relativistic speeds and then returns to Earth at the same speed, the twin that went on the spaceship will be significantly younger than the twin that stayed on Earth.

The apparent paradox is that only one twin experiences a slowing of time when both twins should. Since motion is relative in special relativity, the twin on Earth could be considered to be moving away from the twin in the spaceship at the same speed. If this is true, why doesn’t the twin in the spaceship perceive the twin on Earth to be younger? Why does time dilation only effect the twin in the spaceship when both twins could be considered to be the traveler?

The solution to this paradox is that the situation is not exactly symmetrical for both twins. There are actually three inertial reference frames involved in the problem, not two. One reference frame is for the twin that stays on Earth, the second is for the twin in the spaceship moving away from earth, and the third is for the same twin in the spaceship moving back toward the earth. The twin in the spaceship changes inertial frames when changing direction to begin the return journey. Therefore, since the situation is actually asymmetrical, only the twin in spaceship can be considered the traveler who experiences time dilation.

## General Relativity

The key idea of general relativity is that acceleration due to gravity and uniform acceleration are equivalent. Einstein determined that objects accelerating uniformly under their own power, such as a rocket, and stationary objects under the influence of a gravitational field, such as a person on the surface of the Earth, are actually in the same state of motion. Einstein deduced that free-fall is actually inertial motion (motion with no acceleration).

### Spacetime

In classical mechanics, time is considered constant and does not change based on any event in the three special dimensions. However, in relativity, since the velocity of an object relative to an observer can change the passage of time, time cannot be considered separate from the three spatial dimensions. Spacetime can be thought of as a fabric or surface that masses such as stars and planets sit on, creating distortions in the surface. According to Einstein, the effects of gravity are actually the result of these distortions in spacetime. The term “gravity well” stems from the fact that the distortions caused by the presence of a mass can be visualized as dips in the fabric of spacetime.

### Time Dilation

Time passes at different rates at different gravitational potentials. This effect has been seen in very precise clocks that were taken to different altitudes.

### Gravitational Redshift

This is similar to the Doppler effect for light except that it is caused by a gravitational field. Light emitted from within a strong gravitational field will appear to have a longer wavelength when received by an observer within a weaker gravitational field. Light can also be gravitationally blue shifted by reversing the gravitational fields at the source and the observer.

Relativity cont.

## Important Equations

Lorentz gamma factor (how much dilation or contraction)

$$\gamma = \frac{1}{\sqrt{1- (\frac{y}{c})^2}}$$

γ - Lorentz factor

v - velocity

c - speed of light

Time dilation

$$T^i = \gamma T$$

T - time

Length contraction

$$L^i = \frac{L}{\gamma}$$

L - length

Relativistic mass

$$m = \gamma m_{\circ}$$

m - mass

Potential energy

$$E = mc^2$$

E - potential energy

Speed of light is a constant

$$c = 2.998 × 10^8 m/s$$