Time dilation is a phenomenon where time passes differently between observers. There are two main types of time dilation:

1. Velocity-based time dilation: Time runs slower for objects moving at high speeds
2. Gravitational time dilation: Time runs slower in stronger gravitational fields

Velocity-Based Time Dilation

Velocity-based time dilation occurs when observers are in relative motion. Time appears to run slower for an object moving at high speeds relative to a stationary observer.

Mathematical Description

The relationship between proper time and coordinate time is given by using the Lorentz factor:

Reciprocity

An important aspect of time dilation is that it is reciprocal: each observer sees the other's clock running slow.

Derivation

Inference Based Derivation

Consider a clock designed using light. The clock has two reflective plates and the clock ticks once every time the right reaches plate .

Example diagram of a light clock from both an observer at rest and an observer in motion

The tick rate of the clock from the perspective of the observer in motion can be simply calculated as follows

Hence the clock ticks every seconds

Now the exact same clock but from the perspective of the observer at rest is calculated as follows

Using our previous equation as will remain constant no matter the velocity

We can notice this is the Lorentz factor and re-arranging

we get the original equation defined earlier.