A coordinate system used to measure positions and times of events. As seen in Unit 1, an Inertial Frame is one that is not accelerating (at rest or moving at constant velocity).
This is the "common sense" set of equations for transforming measurements from one inertial frame (S) to another frame (S') that is moving with a constant velocity `v` along the x-axis relative to S.
The last equation, t' = t, is the crucial assumption of classical physics: Time is absolute. Everyone measures the same time interval between two events.
Galilean Invariance (or the Principle of Classical Relativity) states that the laws of mechanics are the same (invariant) in all inertial frames.
If you are in a smooth-moving train (inertial frame), you can play catch, and the ball will behave exactly as it would on the ground. You cannot perform any mechanical experiment *inside* the train to tell if you are moving or at rest.
Problem: Maxwell's equations of electromagnetism (which predict the speed of light `c`) are *not* invariant under Galilean transformations. This implies the laws of E&M are *not* the same in all inertial frames, or that the Galilean transformations are wrong.
In the 19th century, light was known to be a wave. Scientists assumed it *must* travel through a medium, just as sound waves travel through air. They called this invisible, all-pervading medium the "luminiferous ether".
Michelson and Morley used a very sensitive interferometer to detect this tiny difference in the speed of light.
No fringe shift was ever observed.
The experiment was repeated at different times of day and year. The result was always null (zero).
Conclusion: The speed of light is constant in all directions, regardless of the motion of the observer or the source. The idea of the luminiferous ether was wrong.
In 1905, Albert Einstein proposed a new theory based on two simple, radical postulates to explain the Michelson-Morley result and the E&M problem.
"The laws of physics are the same (invariant) in all inertial frames of reference."
This extends Galilean relativity to *all* laws of physics, including electromagnetism. There is no "absolute rest frame"; all inertial frames are equivalent.
"The speed of light in a vacuum (c) has the same value for all inertial observers, regardless of the motion of the source or the observer."
This directly accepts the Michelson-Morley result. It is a radical break from "common sense" and Galilean transformations.
Consequence: To make these postulates work together, the classical idea of absolute time (t' = t) must be abandoned. Time and space are now relative and interwoven into "spacetime".
These are the *new* set of transformation equations that replace the Galilean ones. They are derived directly from Einstein's two postulates. They preserve the laws of E&M and keep `c` constant.
For frame S' moving at velocity `v` along the +x axis relative to frame S:
x' = γ (x - vt)
y' = y
z' = z
t' = γ (t - vx/c²)
Where γ (gamma) is the Lorentz factor:
γ = 1 / √(1 - v²/c²)
Inverse Transformations (from S' to S): Just swap (x, t) with (x', t') and change `v` to `-v`.
x = γ (x' + vt')
t = γ (t' + vx'/c²)
Relativity of Simultaneity: This is a direct consequence of the `t'` equation: `t' = γ (t - vx/c²)`.
Consider two events (A and B) that happen at the same time `t` in frame S, but at different locations `x_A` and `x_B`.
In frame S', the times of the events are:
t'A = γ (t - vxA/c²)
t'B = γ (t - vxB/c²)
The time difference in S' is: Δt' = t'B - t'A = -γv(xB - xA)/c²
Since `x_A ≠ x_B`, the time difference Δt' is not zero.
Events that are simultaneous in one frame (S) are NOT simultaneous in another frame (S') that is moving relative to it.
Order of Events:
A moving object appears shorter in the direction of its motion.
Consider a rod at rest in frame S'. Its "proper length" (length in its rest frame) is L₀ = x'₂ - x'₁.
An observer in frame S wants to measure its length. They must measure the positions of its ends (x₁ and x₂) *at the same time* in their frame (t₁ = t₂ = t).
Using the Lorentz transformation `x' = γ (x - vt)`:
x'₂ = γ (x₂ - vt)
x'₁ = γ (x₁ - vt)
Subtracting the two equations: (x'₂ - x'₁) = γ (x₂ - x₁)
L₀ = γ * L (where L is the length measured in S)
L = L₀ / γ = L₀ √(1 - v²/c²)
Since γ ≥ 1, the measured length L is always less than L₀.
A moving clock runs slow, as measured by a stationary observer.
Consider a clock at rest in frame S'. It ticks at one location (x'₁ = x'₂ = x'). The time interval between two ticks *in its rest frame* is the "proper time", Δt₀ = t'₂ - t'₁.
An observer in S measures this time interval using their own clock. They use the inverse transformation `t = γ (t' + vx'/c²)`:
t₂ = γ (t'₂ + vx'/c²)
t₁ = γ (t'₁ + vx'/c²)
Subtracting the equations: (t₂ - t₁) = γ (t'₂ - t'₁)
Let Δt = t₂ - t₁ (the time interval measured by S, the "moving" observer).
Δt = γ Δt₀ = Δt₀ / √(1 - v²/c²)
Since γ ≥ 1, the measured time interval Δt is always greater than Δt₀.
In words: The observer in S sees the S' clock (which ticks once every Δt₀ seconds) taking a longer time Δt to complete one tick. The moving clock appears to run slow.
This is a thought experiment that highlights the non-intuitive nature of time dilation.
This formula replaces the simple Galilean `v_total = u + v`.
Let frame S see an object moving with velocity `u_x`.
Let frame S' move with velocity `v` relative to S (along x-axis).
What velocity `u_x'` does S' measure for the object?
Derive from Lorentz transformations: `x' = γ(x-vt)` and `t' = γ(t-vx/c²)`.
u_x' = dx'/dt' = [γ(dx - v dt)] / [γ(dt - v dx/c²)] = (dx/dt - v) / (1 - (v/c²)(dx/dt))
u_x' = (u_x - v) / (1 - u_x v / c²)
The inverse (what S sees) is more common:
u_x = (u_x' + v) / (1 + u_x' v / c²)
(Note: The modern concept is that mass `m` is invariant, and relativistic momentum is `p = γmv`. The "relativistic mass" `m_rel = γm` is an older, but still common, way to teach this.)
To conserve momentum in relativity, the definition of momentum must be changed. This is equivalent to saying that mass increases with velocity.
Let `m₀` be the rest mass (mass of the object in its rest frame).
The mass `m(v)` when it is moving at speed `v` is:
m(v) = γ m₀ = m₀ / √(1 - v²/c²)
What about particles that *do* travel at `v = c`, like photons (particles of light)?
If `v = c`, the mass equation is `m(c) = m₀ / √(1 - c²/c²) = m₀ / 0`.
For this to be non-infinite, the *only* possibility is that the rest mass m₀ must be zero.
This is the most famous equation in physics. It is a direct consequence of STR.
Relativistic Kinetic Energy (K):
K = (Work done) = ∫ F dx = ∫ (dp/dt) dx = ... (a complex derivation)
The result is: K = (γ - 1) m₀c²
Total Relativistic Energy (E):
E = K + (Rest Energy)
E = (γ - 1)m₀c² + m₀c²
E = γ m₀c² = m(v) c²
This is the full equation. It combines the kinetic energy and the rest energy.
Rest Energy (E₀):
If the particle is at rest (v=0, γ=1), its total energy is:
E₀ = m₀c²
This is the profound conclusion: Mass is a form of energy. A particle at rest has an "in-built" energy content equal to its rest mass times c².
This explains the energy released in: