# Special relativity

### Inertial reference frame

• Motion of an object is typically described relative to a reference frame
• If the reference frame does not accelerate, and is therefore stopped or moving at a constant speed, it is called an inertial reference
• In this reference frame, Newton’s Law of Inertia holds – things will keep on moving at a constant velocity unless acted upon by a net external force
• Projectile motion is example (e.g. displacement from the origin)

### Newtonian relativity

Postulates:

• All uniform motion is relative and there is no absolute and well defined state of rest
• i.e there is no absolute standard to which everything moves; an absolute inertial reference frame
• All inertia reference frames share a universal time

### Maxwell’s equations

• Derived equations that describe the behaviour of light
• But require speed of light to be constant

Scientists at the time believed that c was only constant relative to a hypothesised medium called the aether, which was theorised to be the absolute unique frame of reference which light needs to travel in.

Experiments failed to detect the aether, and so c must be constant. See book for a bit more detail.

### Special Theory of Relativity

Postulates:

• The principle of relativity. The laws of physics (Newton’s laws) are the same for objects moving in inertial frames of reference
• The principle of the speed of light. The speed of light is the same of all observers, regardless of their motion relative to the source of light.

### Time dilation

Where t = time measured by a stationary observer on Earth

t0 = time measured by an observer at rest relative to the object travelling at v (in the same inertial reference frame)

• The faster the object goes, the smaller t0 gets, relative to t
• The faster the object goes, t would increase or dilate relative to t0
• Both observers would observe that the other’s time has slowed down (or theirs has sped up)

Proper time is the time measured by an observer in their own reference frame.

### Length contraction

Where L is the length of an object measured by a stationary observer on Earth

L0 is the length measured by an observer travelling at rest relative to the object

• As v increases, L decreases relative to L0
• The stationary observer only observes contraction in the direction on motion. All other dimensions stay the same.
• Again, both observers would see length contraction on each other.

### Simultaneity

• Due to the fact that time is different to two different observers, the order of events may be different as well
• What occurs simultaneously to one person, may not be simultaneous to another person moving relative to him

### Relativistic Mass

• As an object approaches the speed of light, it gains mass
• This is due to mass-energy equivalence
• Some of the energy used to accelerate the object converts to mass
• Hence, it would require infinite energy to accelerate an object to the speed of light