The formula that describes time dilation due to relative motion is derived from the principles of special relativity. It relates the proper time (τ) experienced by an object in its own reference frame to the time (t) measured by an observer in a different reference frame. The formula is known as the time dilation equation:
τ = γt
In this equation, γ (gamma) represents the Lorentz factor, which depends on the relative velocity (v) between the two frames of reference. The Lorentz factor is given by:
γ = 1 / √(1 - (v^2/c^2))
Here, c is the speed of light in a vacuum, which is approximately 299,792,458 meters per second.
When the relative velocity (v) is small compared to the speed of light (c), the Lorentz factor approaches 1, and there is negligible time dilation. However, as the relative velocity increases, the Lorentz factor becomes larger, and time dilation becomes more significant. This effect becomes most pronounced as the relative velocity approaches the speed of light.
It's important to note that the time dilation equation describes the time experienced by an object in its own reference frame (proper time) compared to the time measured by an observer in a different reference frame. The equation accounts for the fact that the passage of time is not absolute but depends on the relative motion between observers.
For time dilation due to gravitational fields in the framework of general relativity, the equations become more complex and involve the gravitational potential and the curvature of space-time. The specific form of the equations depends on the specific gravitational field being considered.