Time Transformation: UNIZOR.COM - Relativity 4 All - Einstein View

Notes to a video lecture on UNIZOR.COM

Time Transformation

We have introduced a concept of time dilation in the lecture Time Measuring and, arranging a thought experiment with a light clock, came up with a formula for the perceived time duration of our experiment by an observer outside of a reference frame where this experiment is conducted in terms of the proper time duration measured inside a frame of this experiment.

The formula for time dilation was
T_β = γ·t_β
where
α is some inertial frame with Cartesian coordinates {X,Y,Z}, time T and α-observer in it;
β is another inertial frame with Cartesian coordinates {x,y,z}, time t and β-observer in it;
initially at time T=t=0 α-frame and β-frame coincide;
β-frame is moving with constant speed v relative to α-frame along the X-axis of α, preserving parallelism of corresponding axes;
T_β is a perceived time duration of β-experiment by α-observer;
t_β is the proper time duration of β-experiment as observed by β-observer moving together with β-frame;
γ = 1/√1−(v²/c²);
c is the speed of light in vacuum - a universal constant, the same in all inertial reference frames.

We have also derived the formulas of transformation of Cartesian coordinates and time from one inertial reference frame to another in Lorentz Transformation lecture.
The final form of Lorentz transformation from inertial α-frame coordinates {X,T} to inertial β-frame coordinates {x,t}, when β-frame is uniformly moving with speed v along α's X-axis, in the Special Theory of Relativity is:

x =	X − v·T √1−(v/c)²

t =	T − v·X/c² √1−(v/c)²

If β-frame moves with speed v relative to α-frame along X-axis, we can consider α-frame as moving with speed −v relative to β-frame.
Then the inverse transformation would be

X =	x + v·t √1−(v/c)²

T =	t + v·x/c² √1−(v/c)²

Armed with the above Lorentz Transformation formulas, we can attempt to derive the formula for time dilation more mathematically.
Assume some experiment is conducted at point x=x₀ in β-frame.
The beginning of this experiment is at time t_beg and it ends at time t_end, so the duration of an experiment is
Δt = t_end − t_beg

The α-observer sees this experiment begins at point X_beg and time T_beg, ending at point X_end and time T_end.

According to Lorentz Transformation, the coordinates of β-experiment for α-observer are

Xbeg =

x0 + v·tbeg √1−(v/c)²

Tbeg =

tbeg + v·x0/c² √1−(v/c)²

Xend =

x0 + v·tend √1−(v/c)²

Tend =

tend + v·x0/c² √1−(v/c)²

Now we can calculate the duration of our experiment as perceived by α-observer.
ΔT = T_end − T_beg =

=	tend−tbeg √1−(v/c)²

= Δt·γ
This is exactly the same formula obtained in lecture Time Measuring through a thought experiment with a light clock.

Incidentally, while in β-frame our experiment was conducted at a single point x₀ on x-axis, an observer in α-frame sees it at moving along X-axis for a distance
ΔX = X_end − X_beg =

=	v·(tend − tbeg) √1−(v/c)²

= v·Δt·γ