The formula for time dilation involves the use of the Lorentz factor, which is denoted by the symbol "γ" (gamma). The minus sign in the time dilation formula arises from the nature of spacetime and the spacetime interval.
In special relativity, spacetime is described by a four-dimensional coordinate system that combines space and time into a unified framework. The spacetime interval between two events is a quantity that remains invariant, meaning it does not change, regardless of the observer's frame of reference.
The spacetime interval is given by the formula:
Δs^2 = Δt^2 - Δx^2 - Δy^2 - Δz^2
In this formula, Δt represents the time interval between two events, and Δx, Δy, and Δz represent the differences in the spatial coordinates between those events.
The minus sign in the formula is a result of the signature of the spacetime metric used in special relativity, which is known as the Minkowski metric. The Minkowski metric has a signature of (-, +, +, +), meaning there is a negative sign associated with the time component of the spacetime interval.
When considering time dilation, we are interested in the relationship between the time experienced by an observer in one frame of reference and the time experienced by an observer in a different frame of reference. The Lorentz factor, γ, is used to describe this relationship.
The formula for time dilation is:
Δt' = γ * Δt
Where Δt' is the time experienced by the moving observer, Δt is the time experienced by the stationary observer, and γ is the Lorentz factor, which is given by:
γ = 1 / √(1 - (v^2 / c^2))
In this formula, v represents the relative velocity between the two frames of reference, and c is the speed of light.
The negative sign in the denominator of the Lorentz factor arises from the spacetime interval equation mentioned earlier. It ensures that γ is a real, positive value and allows for time dilation to occur. Without the negative sign, the Lorentz factor would be imaginary, which would not correspond to physical reality.
So, the presence of the minus sign in the time dilation formula is a consequence of the spacetime geometry and the nature of the spacetime interval in special relativity.