Proper time

In relativity, proper time along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval between two events on a world line is the change in proper time. This interval is the quantity of interest, since proper time itself is fixed only up to an arbitrary additive constant, namely the setting of the clock at some event along the world line. The proper time interval between two events depends not only on the events but also the world line connecting them, and hence on the motion of the clock between the events. It is expressed as an integral over the world line. An accelerated clock will measure a smaller elapsed time between two events than that measured by a non-accelerated (inertial) clock between the same two events. The twin paradox is an example of this effect. Δ τ = ∫ P d τ = ∫ d s c . {displaystyle Delta au =int _{P}d au =int {frac {ds}{c}}.} (2) Δ τ = ∫ P 1 c η μ ν d x μ d x ν = ∫ P d t 2 − d x 2 c 2 − d y 2 c 2 − d z 2 c 2 = ∫ 1 − 1 c 2 [ ( d x d t ) 2 + ( d y d t ) 2 + ( d z d t ) 2 ] d t = ∫ 1 − v ( t ) 2 c 2 d t = ∫ d t γ ( t ) , {displaystyle {egin{aligned}Delta au &=int _{P}{frac {1}{c}}{sqrt {eta _{mu u }dx^{mu }dx^{ u }}}\&=int _{P}{sqrt {dt^{2}-{dx^{2} over c^{2}}-{dy^{2} over c^{2}}-{dz^{2} over c^{2}}}}\&=int {sqrt {1-{frac {1}{c^{2}}}left}}dt\&=int {sqrt {1-{frac {v(t)^{2}}{c^{2}}}}}dt=int {frac {dt}{gamma (t)}},end{aligned}}} (3) Δ τ = ∫ P d τ = ∫ P 1 c g μ ν d x μ d x ν . {displaystyle Delta au =int _{P},d au =int _{P}{frac {1}{c}}{sqrt {g_{mu u };dx^{mu };dx^{ u }}}.} (4) In relativity, proper time along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval between two events on a world line is the change in proper time. This interval is the quantity of interest, since proper time itself is fixed only up to an arbitrary additive constant, namely the setting of the clock at some event along the world line. The proper time interval between two events depends not only on the events but also the world line connecting them, and hence on the motion of the clock between the events. It is expressed as an integral over the world line. An accelerated clock will measure a smaller elapsed time between two events than that measured by a non-accelerated (inertial) clock between the same two events. The twin paradox is an example of this effect. In terms of four-dimensional spacetime, proper time is analogous to arc length in three-dimensional (Euclidean) space. By convention, proper time is usually represented by the Greek letter τ (tau) to distinguish it from coordinate time represented by t. By contrast, coordinate time is the time between two events as measured by an observer using that observer's own method of assigning a time to an event. In the special case of an inertial observer in special relativity, the time is measured using the observer's clock and the observer's definition of simultaneity. The concept of proper time was introduced by Hermann Minkowski in 1908, and is a feature of Minkowski diagrams. The formal definition of proper time involves describing the path through spacetime that represents a clock, observer, or test particle, and the metric structure of that spacetime. Proper time is the pseudo-Riemannian arc length of world lines in four-dimensional spacetime. From the mathematical point of view, coordinate time is assumed to be predefined and we require an expression for proper time as a function of coordinate time. From the experimental point of view, proper time is what is measured experimentally and then coordinate time is calculated from the proper time of some inertial clocks. Proper time can only be defined for timelike paths through spacetime which allow for the construction of an accompanying set of physical rulers and clocks. The same formalism for spacelike paths leads to a measurement of proper distance rather than proper time. For lightlike paths, there exists no concept of proper time and it is undefined as the spacetime interval is identically zero. Instead an arbitrary and physically irrelevant affine parameter unrelated to time must be introduced. Let the Minkowski metric be defined by