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# 5: Relativity

• 5.0: Prelude to Relativity
The theory of relativity led to a profound change in the way we perceive space and time. The “common sense” rules that we use to relate space and time measurements in the Newtonian worldview differ seriously from the correct rules at speeds near the speed of light. For example, the special theory of relativity tells us that measurements of length and time intervals are not the same in reference frames moving relative to one another.
• 5.1: Invariance of Physical Laws
Relativity is the study of how observers in different reference frames measure the same event. Modern relativity is divided into two parts. Special relativity deals with observers in uniform (unaccelerated) motion, whereas general relativity includes accelerated relative motion and gravity. Modern relativity is consistent with all empirical evidence thus far and, in the limit of low velocity and weak gravitation, gives close agreement with the predictions of classical (Galilean) relativity.
• 5.2: Relativity of Simultaneity
Two events are defined to be simultaneous if an observer measures them as occurring at the same time (such as by receiving light from the events). Two events at locations a distance apart that are simultaneous for an observer at rest in one frame of reference are not necessarily simultaneous for an observer at rest in a different frame of reference.
• 5.3: Time Dilation
Time dilation is the lengthening of the time interval between two events when seen in a moving inertial frame rather than the rest frame of the events (in which the events occur at the same location). Observers moving at a relative velocity v do not measure the same elapsed time between two events. Proper time Δτ is the time measured in the reference frame where the start and end of the time interval occur at the same location.
• 5.4: Length Contraction
Length contraction is the decrease in observed length of an object from its proper length $$L_0$$ to length L when its length is observed in a reference frame where it is traveling at speed v. The proper length is the longest measurement of any length interval. Any observer who is moving relative to the system being observed measures a length shorter than the proper length.
• 5.5: The Lorentz Transformation
Relativistic phenomena can be explained in terms of the geometrical properties of four-dimensional space-time, in which Lorentz transformations correspond to rotations of axes. The analysis of relativistic phenomena in terms of space-time diagrams supports the conclusion that these phenomena result from properties of space and time itself, rather than from the laws of electromagnetism.
• 5.6: Relativistic Velocity Transformation
Relativistic velocity addition describes the velocities of an object moving at a relativistic velocity. Velocities cannot add to be greater than the speed of light. Although displacements perpendicular to the relative motion are the same in both frames of reference, the time interval between events differ, and differences in dt and dt' lead to different velocities seen from the two frames.
• 5.7: Doppler Effect for Light
An observer of electromagnetic radiation sees relativistic Doppler effects if the source of the radiation is moving relative to the observer. The wavelength of the radiation is longer (called a red shift) than that emitted by the source when the source moves away from the observer and shorter (called a blue shift) when the source moves toward the observer.
• 5.8: Relativistic Momentum
The law of conservation of momentum is valid for relativistic momentum whenever the net external force is zero. The relativistic momentum is $$p = \gamma m u$$, where m is the rest mass of the object, u is its velocity relative to an observer, and the relativistic factor is $$\gamma = \frac{1}{\sqrt{1 - \frac{u^2}{c^2}}}$$.
• 5.9: Relativistic Energy
The rest energy of an object of mass m is $$E_0 = mc^2$$, meaning that mass is a form of energy. If energy is stored in an object, its mass increases. Mass can be destroyed to release energy. Relativistic energy is conserved as long as we define it to include the possibility of mass changing to energy. At extremely high velocities, the rest energy $$mc^2$$ becomes negligible, and $$E = pc$$.