Observers in different reference frames disagree about the time order of two events (including whether the events are simultaneous) if there is not enough time for a signal at light speed to travel from one event to the other.
The Lorentz factor occurs in many relativity equations.
When g is used in expressions for time dilation or length contraction, u in Eq. (26-2) stands for the relative speed of the two reference frames. When g is used in expressions for the momentum, kinetic energy, or total energy of a particle, u in Eq. (26-2) stands for the particle's speed.
In time dilation problems, identify the two events that mark the beginning and end of the time interval in question. The "clock" is whatever measures this time interval. Identify the reference frame in which the clock is at rest. In that frame, the clock measures the proper time interval Dt0. In any other reference frame, the time interval is longer:
In length contraction problems, identify the object whose length is to be measured in two different frames. The length is contracted only in the direction of the object's motion. If the length in question is a distance rather than the length of an actual object, it often helps to imagine the presence of a long measuring stick. Identify the reference frame in which the object is at rest. The length in that frame is the proper length LO. In any other reference frame, the length L is contracted:
Velocities in different reference frames are related by
The subscripts in uBA mean the velocity of B as measured in A's reference frame. Equation (26-5) is written in terms of the components of the three velocities along a straight line. The components are positive for one direction (your choice) and negative for the other. If A moves to the right in B's frame, then B moves to the left in A's frame: uBA = -uAB.
The relativistic expression for momentum is
With relativistic momentum, it is still true that the impulse delivered equals the change in momentum (2.0K) but (1.0K) is not true: the acceleration due to a constant net force gets smaller and smaller as the particle's speed approaches c. Thus, it is impossible to accelerate something to the speed of light.
The rest energy E0 of a particle is its energy as measured in its rest frame. The relationship between rest energy and mass is: