Note that the x ′ x ′ coordinate of both events is the same because the clock is at rest in S ′. The time signal starts as ( x ′, t 1 ′ ) ( x ′, t 1 ′ ) and stops at ( x ′, t 2 ′ ).

Identify the unknown: Δ t = t 2 − t 1.

Use the Lorentz transformation to find the time interval of the signal measured by the communications officer of spaceship S. The captain of S ′ S ′ sends a radio signal that lasts 1.2 s according to that ship’s clock. Spacecraft S ′ S ′ is at rest, eventually heading toward Alpha Centauri, when Spacecraft S passes it at relative speed c/2. Using the Lorentz Transformation for Time Expressing these relations in Cartesian coordinates gives Specifically, the spherical pulse has radius r = c t r = c t at time t in the unprimed frame, and also has radius r ′ = c t ′ r ′ = c t ′ at time t ′ t ′ in the primed frame. The Galilean transformation nevertheless violates Einstein’s postulates, because the velocity equations state that a pulse of light moving with speed c along the x-axis would travel at speed c − v c − v in the other inertial frame. The laws of mechanics are consistent with the first postulate of relativity.

Because the mass is unchanged by the transformation, and distances between points are uncharged, observers in both frames see the same forces F = m a F = m a acting between objects and the same form of Newton’s second and third laws in all inertial frames. Observers in both frames of reference measure the same value of the acceleration. Velocities in each frame differ by the velocity that one frame has as seen from the other frame. We denote the velocity of the particle by u rather than v to avoid confusion with the velocity v of one frame of reference with respect to the other. The relation between the time and coordinates in the two frames of reference is thenĪ x = a x ′, a y = a y ′, a z = a z ′. For simplicity, assume this relative velocity is along the x-axis. Suppose a second frame of reference S ′ S ′ moves with velocity v with respect to the first. As an example, ( x, y, z, t) could denote the position of a particle at time t, and we could be looking at these positions for many different times to follow the motion of the particle. The Galilean Transformation EquationsĪn event is specified by its location and time ( x, y, z, t) relative to one particular inertial frame of reference S. Finally, we examine the resulting Lorentz transformation equations and some of their consequences in terms of four-dimensional space-time diagrams, to support the view that the consequences of special relativity result from the properties of time and space itself, rather than electromagnetism.

Then we examine how this has to be changed to agree with the postulates of relativity. We first examine how position and time coordinates transform between inertial frames according to the view in Newtonian physics. We can gain further insight into how the postulates of relativity change the Newtonian view of time and space by examining the transformation equations that give the space and time coordinates of events in one inertial reference frame in terms of those in another. We have used the postulates of relativity to examine, in particular examples, how observers in different frames of reference measure different values for lengths and the time intervals. Explain the Lorentz transformation and many of the features of relativity in terms of four-dimensional space-time.Derive the corresponding Lorentz transformation equations, which, in contrast to the Galilean transformation, are consistent with special relativity.Describe the Galilean transformation of classical mechanics, relating the position, time, velocities, and accelerations measured in different inertial frames.

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