Special Relativity: Newtonian Relativity

• Study notes on Special Relativity

Newtonian Relativity

• Newton's laws of mechanics are valid in all inertial reference frames.
• Galilean transformation relates coordinates in two inertial frames, S and S'.
• If coordinate origins O and O' coincide at time t=0, and S' has velocity V along x with respect to S, then: r' = r - Vt, which implies x' = x - Vt, y' = y, and z' = z.
• Velocity transformation states v' = v - V
• Acceleration transformation states a' = a
• Newton's second law (F = ma) is valid in all inertial frames.
• Galilean transformation law for velocity is valid for speeds much less than c when using Galilean relativity.
• Experiments with accelerators find particle velocities approaching c, which is inconsistent with F = ma.

Electromagnetic Theory and Luminiferous Ether

• Electromagnetic theory (Maxwell's equations) predicts electromagnetic waves that propagate with the speed of light, c = 1/sqrt(ε₀μ₀).
• A question arose about with respect to which inertial frame the speed of light is measured?
• A theory postulated the existence of luminiferous ether, a medium filling free space through which light propagates at speed c.
• Absolute reference frame is at rest with respect to the ether.
• Since Earth moves in the ether, Galilean transformation implies v' = v - V, where v' is the light's velocity with respect to Earth, v is the light's velocity with respect to the ether, and V is the Earth's velocity with respect to the ether.

Michelson-Morley Experiment (1887)

• Designed to detect differences in the observed light velocity due to Earth's motion through the ether.
• Result of the experiment was negative, indicating no difference (Earth's speed around Sun is approximately 10^4 m/s)

Michelson-Morley Experiment Setup

• Light from a source is split by a half-silvered mirror M₁ and travels along two perpendicular paths.
• Mirrors at the end of each path reflect the light back to the where interference is observed on a screen.
• If superposition occurs a fringe pattern will form and be observed.

Michelson-Morley Experiment Analysis

• The analysis examines the time it takes for light to travel along paths P-M1-P and P-M2-P.
• For path P-M1-P: t₁ = 2l/sqrt(c² - V²)
• For path P-M2-P: t₂ = 2l*c/(c² - V²)
• According to ether theory and Galilean transformation, t₁ - t₂ ≠ 0 and depends on the instrument's orientation with respect to Earth's velocity.

Michelson-Morley Experiment Conclusion

• The experimental result showed no detectable motion of Earth with respect to the ether.
• Fitzgerald and Lorentz proposed in 1892 that if lengths are contracted in the direction of motion by a factor of sqrt(1 - V²/c²), then the null result could be explained for apparatus where l₁ = l₂.

Later Analysis

• The length contraction hypothesis could explain null results only when the lengths of apparatus arms were equal.
• Does not account for unequal components l₁≠l₂ as was later tested for in the Kennedy-Thorndike experiment.
• Einstein proposed that the speed of light c in a vacuum is the same in all inertial frames of reference.
• Einstein's proposal is inconsistent with the Galilean velocity transformation which required it's alteration including ideas about absolute measurements of length and time.

Einstein's Postulates of Special Relativity

• Postulate 1 (Principle of Relativity): All inertial frames of reference are equivalent with respect to the laws of physics. This means the laws of physics are the same in all inertial frames and includes mechanics and electromagnetic theory.
• Postulate 2: The speed of light in a vacuum is c in all inertial frames of reference.
• Postulate 1 is an extension of Newtonian relativity to include electromagnetism.
• Postulate 2 derives from Postulate 1, explains the Michelson-Morley experiment, and contradicts the notion of absolute time measurements.

Simultaneity and Measurement of Time

• The motion r(t) of an object must be observed.
• Observation of time t at which an object is at r: According to Einsten "train arrives at 700" means the watch reading 700 and the train arrival are simultaneous events, which are spatialy colocal (both occur at r).
• To define simultaneity of events at different locations r₁, r₂, synchronize a clock at r₁ with another clock at r₂ with clocks at rest in the same intertial frame.

Time Measurement Example

• Clocks at A and B, at rest in frame S and synchronized.
• A light signal is sent from A at t=0, observer A observes its reflected signal at t = t₀.
• The light signal reaches B at t = t₀/2.
• B sets his clock to t₀/2 when the light signal is observed.
• Simultaneity of distant events is based on the calculation to account for the signals flight.
• Clock synchronization with A and B depends on using the same inertial frame of reference.
• 3 observers A, B, and Care at rest in inertial frame S and with equal distances separating each (xAB = XBC)
• B sends out radio signals at t=0.
• A and C receive the signals at the same time t₀, as seen in S and are considered simultaneous

Minkowski Diagram

• "World line" is a line in space time that indicates the trajectory of a particle.
• Events are defined as space-time points.
• If frame S' is moving with speed V with respect to S, an observer in S' will see the same events differently.
• An inertial frame of reference can observe the distances traveled by each photon but can derive the speed at which the photons are traveling.
• Cannot infer that relative velocity V+c represents speed which A sees the light, A is at rest in S not S'.
• Observer at S' sees A at a different time than C, the events A and C cannot happen simultaneously.
• Absolute time is relative.
• Light speed is constant in all inertial frames.

Study notes on Newtonian Relativity and Galilean transformations. Includes velocity and acceleration transformations and the validity of Newton's second law in inertial frames. Also covers electromagnetic theory and luminiferous ether.