Fundamentals of Time and Relativity

Lorentz Boost

  • In relativity, a transformation of spacetime coordinates from one inertial frame to another is called a Lorentz transformation.
  • A Lorentz boost, visualized in a Minkowski diagram, amounts to the symmetric rotation of the time- and space axes towards each other.

Consider two inertial systems, one stationary and a second one ‘boosted’, moving with velocity v . We work out the transformation of the spacetime coordinates for a Lorentz boost in the positive x -direction, from the spacetime coordinates {t,x} , associated with an observer O in the stationary inertial frame, to the coordinates { t ′ , x ′ } set up by an observer O ′ in the moving (boosted) inertial frame. We may assume that the coordinates y= y ′ ,z= z ′ are not affected by the transformation. For simplicity, we also assume that at times t= t ' =0 the origins of both systems coincide.

A useful way is to visualize the effect of the transformation in a 2-dimensional Minkowski diagram, see Fig.5.1 below. In this diagram we draw the second inertial frame of observers moving with constant velocity v to the right (red arrows).

Three worldlines
Fig.5.1 Two moving observers (red arrows) synchronize their clocks (red dotted lines)

The red observers synchronize their clocks according to Einstein’s prescription. This means that in the moving frame their clocks are synchronized if

5.1

t B ′ = 1 2 ( t ¯ A ′ + t A ′ )= t A ′ + 1 2 ( t ¯ A ′ − t A ′ )

The worldline of the ray of light in Fig.5.1 forms a right-angled triangle with the red arrow from the origin. This implies:

  1. the time t B ′ is indeed halfway between t ¯ A ′ and t A ′ ;
  2. the red dotted lines are lines of synchrony in the red frame, and therefore the new spatial coordinate axes;
  3. the length AB ¯ ′ is half the length ( t ¯ A ′ − t A ′ ) ;
  4.  the first stretch of the worldline of the ray of light is the bisector in both frames of reference.

Because the 'rotation' of the coordinate axes is symmetric, the velocity of the light ray has been invariant under this transformation, as is the requirement under any Lorentz transformation.

  • The two directions e 0 +v e 1 and e 1 +v e 0 of the new time and space coordinate axes are Minkowski orthogonal.
  • Fig.5.1 shows that space and time cannot be separated in an objective way, so eloquently worded by Minkowski: “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, .. “.