Special Relativity Postulates and Lorentz Transformations

Manual: General · Subject: Physics

Derives the Lorentz transformation and the kinematic structure of special relativity from Einstein’s postulates.

Einstein’s Two Postulates

Postulate 1

The laws of physics are identical in all inertial frames. This is a symmetry statement: no inertial frame is preferred for the formulation of fundamental law.

Postulate 2

The speed of light in vacuum is the same for all inertial observers, independent of the motion of the source or observer. This empirical input forces a replacement of Galilean kinematics.

Lorentz Transformation

For relative speed $v$ along the $x$ -axis, the standard Lorentz transformation is $x' = \gamma(x-vt),\quad t' = \gamma\left(t-\frac{vx}{c^2}\right),\quad y'=y,\quad z'=z,$ where $\gamma = \frac{1}{\sqrt{1-v^2/c^2}}.$

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Interpretive Insight

Lorentz transformations are not merely coordinate tricks; they encode the geometry of spacetime itself.

Which quantity remains invariant under Lorentz transformations in flat spacetime?

The Newtonian time interval

\Delta t

The spacetime interval

s^2 = c^2\Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2

The individual coordinate time

t

in every frame The ordinary 3-velocity of every particle

Write the Lorentz factor $\gamma$ as a function of $v$ and $c$ .

Derivation Strategy

1

Step 1: Assume homogeneity and isotropy so transformations are linear.
2

Step 2: Impose the invariance of light propagation $x = ct$ .
3

Step 3: Solve for coefficients consistent with reciprocity between inertial frames.
4

Step 4: Identify the resulting velocity-dependent scale factor as $\gamma$ .

Galilean vs Lorentzian Kinematics

Galilean

Time is universal: $t'=t$
Velocity addition is linear
Fails to preserve light speed

Lorentzian

Time mixes with space: $t' = \gamma(t-vx/c^2)$
Velocity addition is nonlinear
Preserves $c$ and the spacetime interval

In the standard Lorentz transformation, why does time depend on position?

Because clocks physically stop near moving rods Because simultaneity is frame-dependent Because gravity is always present Because the transformation is non-linear in space

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