Relativity Physics Lesson 38 by Owen Borville 1.15.2026
Relativity is the study of how different observers describe and measure the same event. Modern relativity includes special relativity, which describes observers who are in uniform (unaccelerated) motion, and general relativity, which describes accelerated relative motion and gravity. Modern relativity is empirically accurate and with low velocity and weak gravitation gives the same predictions as classical (Galilean) relativity.
An inertial frame of reference is a reference frame in which a body at rest remains at rest and a body in motion moves at constant speed in a straight line unless acted on by an outside force.
Modern relativity is based on Albert Einstein's two postulates: (1) special relativity postulates that the idea that the laws of physics are the same and can be stated in their simplest form in all inertial frames of reference. (2) special relativity postulates that the speed of light c is a constant in all inertial frames of reference, independent of the relative motion of the source and observer. The Michelson-Morley experiment demonstrated that the speed of light in a vacuum is independent of the motion of the Earth about the Sun.
Two simultaneous events occur if an observer measures them occurring at the same time. These two events are not necessarily simultaneous to all observers in different reference frames and simultaneity is not absolute.
Time dilation is the phenomenon of time passing slower for an observer who is moving relative to another observer (rather than a rest frame). Observers moving at a relative velocity v do not measure the same elapsed time for an event or between two events. Proper time Δt0 is the time measured by an observer at rest relative to the event being observed (event at same location). Proper time is related to the time Δt measured by an Earth-bound observer in the equation:
Δt = Δt0/√1-v^2/c^2 = γΔt0 (τ (tau) can be used for proper time instead of to )
γ(gamma) = 1/√1-v^2/c^2 (Lorentz factor)
The equation relating proper time and time measured by an Earth-bound observer implies that relative velocity cannot exceed the speed of light. The twin paradox asks why a twin travelling at a relativistic speed away and then back towards the Earth ages less than the Earth-bound twin. The premise to the paradox is faulty because the travelling twin is accelerating. Special relativity does not apply to accelerating frames of reference and the journey is not symmetrical for the two twins. Time dilation is usually negligible at low relative velocities, but it does occur, and it has been verified by experiment.
The proper time is the shortest measure of any time interval and any observer who is moving relative to the system being observed measures a time interval longer than the proper time.
All observers agree upon relative speed. Distance depends on an observer's motion. Proper length L0 is the distance between two points measured by an observer who is at rest relative to both of the points. Earth-bound observers measure proper length when measuring the distance between two points that are stationary relative to the Earth. 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.
Length contraction L is the shortening of the measured length of an object moving relative to the observer's frame: L = L0√1-v^2/c^2 = L0/γ Length contraction is the decrease in observed length of an object from its proper length Lo to length L when its length is observed in a reference frame where it is traveling at speed v.
Classical velocity addition velocities add like regular numbers in one dimensional motion: u = v + u', where v is the velocity between two observers, u is the velocity of an object relative to one observer, and u' is the velocity relative to the other observer. Velocities cannot add to be greater than the speed of light.
Relativistic velocity addition describes the velocities of an object moving at a relativistic speed: u = (v + u')/(1+vu'/c^2)
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 red shift) than that emitted by the source when the source moves away from the observer and shorter (called blue shift) when the source moves toward the observer. The shifted wavelength is described by the equation:
λobs = λs √(1 + u/c)/(1-u/c), where λobs is the observed wavelength, λs is the source wavelength, and u is the relative velocity of the source to the observer.
The relativistic Doppler effect for frequency, fobs = fs√(1 - u/c)/(1 + u/c)
The law of conservation of momentum is valid for relativistic momentum whenever the net external force is zero.
Relativistic momentum p is classical momentum multiplied by the relativistic factor γ. Therefore, p = γmu, where m is the rest mass of the object, u is its velocity relative to an observer, and the relativistic factor γ = 1/(√1-u^2/c^2)
At low velocities, relativistic momentum is equivalent to classical momentum. Relativistic momentum approaches infinity as u approaches c. This implies that an object with mass cannot reach the speed of light. Relativistic momentum is conserved, just as classical momentum is conserved.
Relativistic energy is conserved as long as it includes the possibility of mass changing to energy. Total energy of a particle with mass m travelling at speed u is = E = γmc^2, where γ = 1/(√1-v^2/c^2)
Rest energy = E0 = 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. However, the increase or decrease in mass of an object is usually not noticed because the change in mass is so small for a large increase in energy.
Relativistic work-energy theorem is Wnet = E - E0 = γ mc^2 - mc^2 = (γ - 1) mc^2
Relativistically, Wnet = KErel, where KErel is the relativistic kinetic energy.
Relativistic kinetic energy is KErel = (γ -1) mc^2, where γ = 1/(√1-v^2/c^2). At low velocities, relativistic kinetic energy reduces to classical kinetic energy.
No object with mass can reach the speed of light because an infinite amount of work and an infinite amount of energy input is required to accelerate a mass to the speed of light.
The equation E^2 = (pc)^2 + (mc^2)^2 relates the relativistic total energy E and the relativistic momentum p. At extremely high velocities, the rest energy mc^2 becomes negligible, and E = pc.
The Galilean transformation equations describe how, in classical nonrelativistic mechanics, the position, velocity, and accelerations measured in one frame appear in another. Lengths remain unchanged and a single universal time scale is assumed to apply to all internal frames. Newton's laws of mechanics obey the principle of having the same form in all inertial frames under a Galilean transformation: x = x' + vt, y = y', z = z', t = t'. The concept that times and distances are the same in all internal frames in the Galilean transformation, however, is inconsistent with the postulates of special relativity.
Relativistically correct Lorentz transformation equations:
t = (t' + vx'/c^2)/(√1-v^2/c^2)
x = (x'+vt')/(√1-v^2/c^2)
y = y'
z = z'
Relativistically correct Inverse Lorentz transformation equations:
t' = (t-vx/c^2)/(√1-v^2/c^2)
x' = (x-vt)/(√1-v^2/c^2)
y' = y
z' = z
These equations are obtained by requiring an expanding spherical light signal to have the same shape and speed of growth, c, in both reference frames.
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 Lorentz transformation corresponds to a space-time axis rotation, similar in some ways to a rotation of space axes, but in which the invariant spatial separation is given by Δs rather than distances Δr, and that the Lorentz transformation involving the time axis does not preserve perpendicularity of axes or the scales along the axes.
The analysis of relativistic phenomena in terms of space-time diagrams supports the conclusion that these phenomena result from properties of space-time itself, rather than from the laws of electromagnetism.
Relativity is the study of how different observers describe and measure the same event. Modern relativity includes special relativity, which describes observers who are in uniform (unaccelerated) motion, and general relativity, which describes accelerated relative motion and gravity. Modern relativity is empirically accurate and with low velocity and weak gravitation gives the same predictions as classical (Galilean) relativity.
An inertial frame of reference is a reference frame in which a body at rest remains at rest and a body in motion moves at constant speed in a straight line unless acted on by an outside force.
Modern relativity is based on Albert Einstein's two postulates: (1) special relativity postulates that the idea that the laws of physics are the same and can be stated in their simplest form in all inertial frames of reference. (2) special relativity postulates that the speed of light c is a constant in all inertial frames of reference, independent of the relative motion of the source and observer. The Michelson-Morley experiment demonstrated that the speed of light in a vacuum is independent of the motion of the Earth about the Sun.
Two simultaneous events occur if an observer measures them occurring at the same time. These two events are not necessarily simultaneous to all observers in different reference frames and simultaneity is not absolute.
Time dilation is the phenomenon of time passing slower for an observer who is moving relative to another observer (rather than a rest frame). Observers moving at a relative velocity v do not measure the same elapsed time for an event or between two events. Proper time Δt0 is the time measured by an observer at rest relative to the event being observed (event at same location). Proper time is related to the time Δt measured by an Earth-bound observer in the equation:
Δt = Δt0/√1-v^2/c^2 = γΔt0 (τ (tau) can be used for proper time instead of to )
γ(gamma) = 1/√1-v^2/c^2 (Lorentz factor)
The equation relating proper time and time measured by an Earth-bound observer implies that relative velocity cannot exceed the speed of light. The twin paradox asks why a twin travelling at a relativistic speed away and then back towards the Earth ages less than the Earth-bound twin. The premise to the paradox is faulty because the travelling twin is accelerating. Special relativity does not apply to accelerating frames of reference and the journey is not symmetrical for the two twins. Time dilation is usually negligible at low relative velocities, but it does occur, and it has been verified by experiment.
The proper time is the shortest measure of any time interval and any observer who is moving relative to the system being observed measures a time interval longer than the proper time.
All observers agree upon relative speed. Distance depends on an observer's motion. Proper length L0 is the distance between two points measured by an observer who is at rest relative to both of the points. Earth-bound observers measure proper length when measuring the distance between two points that are stationary relative to the Earth. 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.
Length contraction L is the shortening of the measured length of an object moving relative to the observer's frame: L = L0√1-v^2/c^2 = L0/γ Length contraction is the decrease in observed length of an object from its proper length Lo to length L when its length is observed in a reference frame where it is traveling at speed v.
Classical velocity addition velocities add like regular numbers in one dimensional motion: u = v + u', where v is the velocity between two observers, u is the velocity of an object relative to one observer, and u' is the velocity relative to the other observer. Velocities cannot add to be greater than the speed of light.
Relativistic velocity addition describes the velocities of an object moving at a relativistic speed: u = (v + u')/(1+vu'/c^2)
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 red shift) than that emitted by the source when the source moves away from the observer and shorter (called blue shift) when the source moves toward the observer. The shifted wavelength is described by the equation:
λobs = λs √(1 + u/c)/(1-u/c), where λobs is the observed wavelength, λs is the source wavelength, and u is the relative velocity of the source to the observer.
The relativistic Doppler effect for frequency, fobs = fs√(1 - u/c)/(1 + u/c)
The law of conservation of momentum is valid for relativistic momentum whenever the net external force is zero.
Relativistic momentum p is classical momentum multiplied by the relativistic factor γ. Therefore, p = γmu, where m is the rest mass of the object, u is its velocity relative to an observer, and the relativistic factor γ = 1/(√1-u^2/c^2)
At low velocities, relativistic momentum is equivalent to classical momentum. Relativistic momentum approaches infinity as u approaches c. This implies that an object with mass cannot reach the speed of light. Relativistic momentum is conserved, just as classical momentum is conserved.
Relativistic energy is conserved as long as it includes the possibility of mass changing to energy. Total energy of a particle with mass m travelling at speed u is = E = γmc^2, where γ = 1/(√1-v^2/c^2)
Rest energy = E0 = 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. However, the increase or decrease in mass of an object is usually not noticed because the change in mass is so small for a large increase in energy.
Relativistic work-energy theorem is Wnet = E - E0 = γ mc^2 - mc^2 = (γ - 1) mc^2
Relativistically, Wnet = KErel, where KErel is the relativistic kinetic energy.
Relativistic kinetic energy is KErel = (γ -1) mc^2, where γ = 1/(√1-v^2/c^2). At low velocities, relativistic kinetic energy reduces to classical kinetic energy.
No object with mass can reach the speed of light because an infinite amount of work and an infinite amount of energy input is required to accelerate a mass to the speed of light.
The equation E^2 = (pc)^2 + (mc^2)^2 relates the relativistic total energy E and the relativistic momentum p. At extremely high velocities, the rest energy mc^2 becomes negligible, and E = pc.
The Galilean transformation equations describe how, in classical nonrelativistic mechanics, the position, velocity, and accelerations measured in one frame appear in another. Lengths remain unchanged and a single universal time scale is assumed to apply to all internal frames. Newton's laws of mechanics obey the principle of having the same form in all inertial frames under a Galilean transformation: x = x' + vt, y = y', z = z', t = t'. The concept that times and distances are the same in all internal frames in the Galilean transformation, however, is inconsistent with the postulates of special relativity.
Relativistically correct Lorentz transformation equations:
t = (t' + vx'/c^2)/(√1-v^2/c^2)
x = (x'+vt')/(√1-v^2/c^2)
y = y'
z = z'
Relativistically correct Inverse Lorentz transformation equations:
t' = (t-vx/c^2)/(√1-v^2/c^2)
x' = (x-vt)/(√1-v^2/c^2)
y' = y
z' = z
These equations are obtained by requiring an expanding spherical light signal to have the same shape and speed of growth, c, in both reference frames.
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 Lorentz transformation corresponds to a space-time axis rotation, similar in some ways to a rotation of space axes, but in which the invariant spatial separation is given by Δs rather than distances Δr, and that the Lorentz transformation involving the time axis does not preserve perpendicularity of axes or the scales along the axes.
The analysis of relativistic phenomena in terms of space-time diagrams supports the conclusion that these phenomena result from properties of space-time itself, rather than from the laws of electromagnetism.