Special Theory of Relativity: Complete Guide to Einstein's Relativity, Lorentz Transformations & E=mc²

House of Physics October 18, 2025
Special Theory of Relativity: Complete Physics Guide | Einstein's Relativity

Special Theory of Relativity: Complete Physics Guide

Mastering Einstein's Relativity: Lorentz Transformations, Time Dilation, Length Contraction, and E=mc²
Special Theory of Relativity Lorentz Transformation Time Dilation Length Contraction E=mc² Reading Time: 25 min

📋 Table of Contents

📜 Historical Background

The Special Theory of Relativity was developed by Albert Einstein in 1905, revolutionizing our understanding of space, time, and motion:

  • 1905 - "Annus Mirabilis": Einstein published four groundbreaking papers, including "On the Electrodynamics of Moving Bodies" introducing special relativity
  • Background: Attempts to detect the "luminiferous ether" through experiments like Michelson-Morley (1887) had failed
  • Key Insight: Einstein realized that the speed of light is constant for all observers, regardless of their motion
  • Impact: Revolutionized physics, leading to new understanding of space-time, energy-mass equivalence, and eventually general relativity

Einstein's theory resolved contradictions between Maxwell's electromagnetism and Newtonian mechanics, establishing a new framework for physics.

Introduction to Special Relativity

🔬 Frame of Reference

A coordinate system relative to which measurements are taken is known as a frame of reference. There are two main types:

  • Inertial Frame of Reference: A coordinate system in which Newton's first law of motion (law of inertia) is valid. These frames are either at rest or moving with constant velocity.
  • Non-Inertial Frame of Reference: An accelerated frame of reference where fictitious forces appear.

The Special Theory of Relativity deals exclusively with inertial frames of reference.

📝 Relativistic Effects

The Theory of Relativity shows the effects of relative motion on physical quantities. These effects become significant at relativistic speeds:

\[ \text{Relativistic Speed} \geq \frac{c}{10} \]

where \(c\) is the speed of light (\(3 \times 10^8\) m/s). At these high velocities, classical Newtonian mechanics breaks down, and relativistic effects must be considered.

Postulates of Special Relativity

📜 Einstein's Two Postulates

In 1905, Albert Einstein formulated his special theory of relativity based on two fundamental postulates:

1. Principle of Relativity

🌍 First Postulate

"The laws of Physics have the same form in all frames of reference moving with constant velocities with respect to one another."

This can also be stated as: "The laws of Physics are invariant under transformation between all inertial frames."

💡 Implications

The first postulate shows that laws of Physics are absolute and universal and are the same for all inertial observers. So the laws of Physics that hold for one inertial observer cannot be violated for any other inertial observer.

This means there is no preferred or absolute frame of reference in the universe. All inertial frames are equivalent for describing physical phenomena.

2. Principle of Constancy of Speed of Light

💡 Second Postulate

"The speed of light in free space has the same value for all observers regardless of their state of motion."

This can also be stated as: "The speed of light in free space has the same value 'c' in all inertial frames of references."

🧪 Experimental Verification

To understand the second postulate, consider three observers A, B and C at rest in three different inertial frames:

Observer Setup
[Diagram: Three observers with light flash]
  • A flash of light emitted by observer A is observed by him to travel at speed c
  • If frame B is moving away from A at c/4, Galilean kinematics predicts B measures speed: \(c - c/4 = 3c/4\)
  • If frame C is moving toward A at c/4, Galilean transformation predicts C measures speed: \(c + c/4 = 5c/4\)

However, according to the second postulate, all three observers measure the same speed c for the flash of light.

Experimental Evidence

In 1964, a proton accelerator produced a beam of neutral pions (π⁰) which rapidly decay into gamma rays:

\[ \pi^0 \rightarrow \gamma + \gamma \]

The speed of moving pions was measured at 0.99975c. According to Galileo, the gamma rays emitted in the direction of motion should have speed \(c + 0.99975c\), but the measured speed was exactly c, consistent with Einstein's second postulate.

Galilean vs Lorentz Transformations

Galilean Transformation

📐 Classical Transformation

Consider two observers in two different inertial frames S and S'. Frame S is at rest and S' is moving with uniform velocity v along the x-axis with respect to frame S. Suppose at t = 0, the origins of two frames coincide.

Both observers observe the same event. The position and time of the event observed by S is (x, y, z, t) and by S' is (x', y', z', t'). According to Galilean Transformation:

\[ \begin{cases} x' = x - vt \\ y' = y \\ z' = z \\ t' = t \end{cases} \]
Fundamental Equation of Special Relativity

Consider a wave of light that starts from O and O' at t = 0 with speed c. The wave reaches point P after time t from O and takes time t' to reach P from O'.

Distance from O to P:
\[ |OP| = ct \]
\[ \Rightarrow \sqrt{x^2 + y^2 + z^2} = ct \]
\[ \Rightarrow x^2 + y^2 + z^2 = c^2t^2 \]
\[ \Rightarrow x^2 + y^2 + z^2 - c^2t^2 = 0 \quad \text{(1)} \]
Distance from O' to P:
\[ |O'P| = ct' \]
\[ \Rightarrow \sqrt{x'^2 + y'^2 + z'^2} = ct' \]
\[ \Rightarrow x'^2 + y'^2 + z'^2 = c^2t'^2 \]
\[ \Rightarrow x'^2 + y'^2 + z'^2 - c^2t'^2 = 0 \quad \text{(2)} \]
Comparing equations (1) and (2):
\[ x^2 + y^2 + z^2 - c^2t^2 = x'^2 + y'^2 + z'^2 - c^2t'^2 \quad \text{(3)} \]

This is the fundamental equation of special theory of relativity given by Einstein in 1905.

⚠️ Failure of Galilean Transformation

Applying Galilean transformation values to the fundamental equation of relativity:

\[ x^2 + y^2 + z^2 - c^2t^2 = (x - vt)^2 + y^2 + z^2 - c^2t^2 \]
\[ = x^2 - 2xvt + v^2t^2 + y^2 + z^2 - c^2t^2 \]

This does not satisfy the fundamental equation (3), showing that Galilean transformation is incompatible with the constancy of the speed of light.

Lorentz Transformation

📐 Relativistic Transformation

The Lorentz transformation is the correct transformation that satisfies both postulates of special relativity:

\[ \begin{cases} x' = \gamma(x - vt) \\ y' = y \\ z' = z \\ t' = \gamma\left(t - \frac{vx}{c^2}\right) \end{cases} \]

where \(\gamma\) is the Lorentz factor:

\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]

The inverse Lorentz transformation is:

\[ \begin{cases} x = \gamma(x' + vt') \\ y = y' \\ z = z' \\ t = \gamma\left(t' + \frac{vx'}{c^2}\right) \end{cases} \]

Derivation of Lorentz Transformation

🧮 Derivation of Lorentz Transformation

Step 1: Linear Transformation Assumption

We assume the transformation between frames is linear:

\[ x' = a_{11}x + a_{12}y + a_{13}z + a_{14}t \]
\[ y' = a_{21}x + a_{22}y + a_{23}z + a_{24}t \]
\[ z' = a_{31}x + a_{32}y + a_{33}z + a_{34}t \]
\[ t' = a_{41}x + a_{42}y + a_{43}z + a_{44}t \]

Step 2: Simplification Based on Setup

Since relative motion is along x-axis only:

\[ y' = y \]
\[ z' = z \]

Also, the transformation must be symmetric in y and z directions:

\[ x' = a_{11}x + a_{14}t \]
\[ t' = a_{41}x + a_{44}t \]

Step 3: Apply Fundamental Equation

\[ x^2 + y^2 + z^2 - c^2t^2 = x'^2 + y'^2 + z'^2 - c^2t'^2 \]
\[ \Rightarrow x^2 - c^2t^2 = (a_{11}x + a_{14}t)^2 - c^2(a_{41}x + a_{44}t)^2 \]

Step 4: Coefficient Comparison

\[ x^2 - c^2t^2 = (a_{11}^2 - c^2a_{41}^2)x^2 + (2a_{11}a_{14} - 2c^2a_{41}a_{44})xt + (a_{14}^2 - c^2a_{44}^2)t^2 \]

Comparing coefficients:

\[ a_{11}^2 - c^2a_{41}^2 = 1 \quad \text{(4)} \]
\[ a_{14}^2 - c^2a_{44}^2 = -c^2 \quad \text{(5)} \]
\[ 2a_{11}a_{14} - 2c^2a_{41}a_{44} = 0 \quad \text{(6)} \]

Step 5: Boundary Conditions

Origin of S' frame: \(x' = 0\) corresponds to \(x = vt\)

\[ 0 = a_{11}(vt) + a_{14}t \]
\[ \Rightarrow a_{14} = -va_{11} \quad \text{(7)} \]

Step 6: Solve the System

From equations (4), (5), (6), and (7), we can solve for the coefficients:

\[ a_{11} = \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]
\[ a_{14} = -\gamma v \]
\[ a_{41} = -\frac{\gamma v}{c^2} \]
\[ a_{44} = \gamma \]

Step 7: Final Transformation

\[ x' = \gamma(x - vt) \]
\[ t' = \gamma\left(t - \frac{vx}{c^2}\right) \]

This is the Lorentz transformation.

Consequences of Special Relativity

Time Dilation

⏱️ Time Dilation

Time dilation is the phenomenon where time appears to pass slower for objects moving relative to an observer. The time interval between two events is shortest in the frame where the events occur at the same position (proper time).

🧮 Derivation of Time Dilation

Step 1: Proper Time

Consider a clock at rest in frame S' at position \(x' = 0\). The time interval between two ticks measured in S' is \(\Delta t' = t_2' - t_1'\). This is the proper time.

Step 2: Transformation to Frame S

Using the Lorentz transformation:

\[ t = \gamma\left(t' + \frac{vx'}{c^2}\right) \]

Since \(x' = 0\):

\[ t = \gamma t' \]
\[ \Rightarrow \Delta t = \gamma \Delta t' \]
Step 3: Time Dilation Formula
\[ \Delta t = \frac{\Delta t'}{\sqrt{1 - \frac{v^2}{c^2}}} \]

Since \(\gamma > 1\), \(\Delta t > \Delta t'\), meaning time appears to run slower in the moving frame.

Sample Problem: Time Dilation

A spaceship travels at 0.99c relative to Earth. If 1 hour passes on the spaceship, how much time passes on Earth?

Given:
\[ v = 0.99c \]
\[ \Delta t' = 1 \, \text{hour} \]
Lorentz factor:
\[ \gamma = \frac{1}{\sqrt{1 - (0.99)^2}} \]
\[ = \frac{1}{\sqrt{1 - 0.9801}} \]
\[ = \frac{1}{\sqrt{0.0199}} \]
\[ = \frac{1}{0.141067} \]
\[ \approx 7.088 \]
Time on Earth:
\[ \Delta t = \gamma \Delta t' \]
\[ = 7.088 \times 1 \]
\[ = 7.088 \, \text{hours} \]

Length Contraction

📏 Length Contraction

Length contraction is the phenomenon where the length of an object moving relative to an observer appears shorter in the direction of motion. The length is longest in the frame where the object is at rest (proper length).

🧮 Derivation of Length Contraction

Step 1: Proper Length

Consider a rod at rest in frame S' with ends at \(x_1'\) and \(x_2'\). The proper length is \(L_0 = x_2' - x_1'\).

Step 2: Measurement in Frame S

To measure the length in frame S, we need the positions \(x_1\) and \(x_2\) at the same time \(t\). Using Lorentz transformation:

\[ x' = \gamma(x - vt) \]
\[ \Rightarrow x_1' = \gamma(x_1 - vt) \]
\[ x_2' = \gamma(x_2 - vt) \]
Step 3: Length Contraction Formula
\[ L_0 = x_2' - x_1' = \gamma(x_2 - x_1) \]
\[ \Rightarrow L = x_2 - x_1 = \frac{L_0}{\gamma} \]
\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]

Since \(\gamma > 1\), \(L < L_0\), meaning the length appears shorter in the moving frame.

Sample Problem: Length Contraction

A spaceship with proper length 100 m travels at 0.87c relative to Earth. What is its length as measured by an Earth observer?

Given:
\[ L_0 = 100 \, \text{m} \]
\[ v = 0.87c \]
Length contraction factor:
\[ \sqrt{1 - \frac{v^2}{c^2}} = \sqrt{1 - (0.87)^2} \]
\[ = \sqrt{1 - 0.7569} \]
\[ = \sqrt{0.2431} \]
\[ = 0.493 \]
Length as measured from Earth:
\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]
\[ = 100 \times 0.493 \]
\[ = 49.3 \, \text{m} \]

Relativistic Mass

⚖️ Relativistic Mass

According to special relativity, the mass of an object increases with its velocity relative to an observer:

\[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}} \]

where \(m_0\) is the rest mass (mass when the object is at rest) and \(m\) is the relativistic mass.

💡 Implications

As an object approaches the speed of light, its relativistic mass approaches infinity. This explains why no object with mass can reach or exceed the speed of light - it would require infinite energy to accelerate an object with infinite mass.

Mass-Energy Equivalence

⚡ Einstein's Famous Equation

Perhaps the most famous equation in all of physics, Einstein's mass-energy equivalence states:

\[ E = mc^2 \]

where \(E\) is the total energy of the object, \(m\) is its relativistic mass, and \(c\) is the speed of light.

🧮 Derivation of E=mc²

Step 1: Relativistic Momentum
\[ p = mv = \frac{m_0 v}{\sqrt{1 - \frac{v^2}{c^2}}} \]
Step 2: Work-Energy Theorem

The work done on an object equals its change in kinetic energy:

\[ K = \int F \, dx = \int \frac{dp}{dt} dx = \int v \, dp \]
Step 3: Integration
\[ K = \int_0^p v \, dp = \int_0^v v \, d\left(\frac{m_0 v}{\sqrt{1 - \frac{v^2}{c^2}}}\right) \]
\[ = m_0 \int_0^v v \, d\left(\frac{v}{\sqrt{1 - \frac{v^2}{c^2}}}\right) \]
Step 4: Evaluate Integral
\[ K = m_0 \int_0^v \left( \frac{v}{(1 - v^2/c^2)^{3/2}} \right) dv \]
\[ = m_0 c^2 \left[ \frac{1}{\sqrt{1 - v^2/c^2}} - 1 \right] \]
\[ = mc^2 - m_0 c^2 \]
Step 5: Total Energy
\[ E = K + m_0 c^2 = mc^2 \]

This shows that the total energy of an object is equal to its relativistic mass times \(c^2\).

Sample Problem: Mass-Energy Equivalence

Calculate the energy equivalent of 1 gram of matter.

Given:
\[ m = 1 \, \text{g} = 0.001 \, \text{kg} \]
\[ c = 3 \times 10^8 \, \text{m/s} \]
Energy equivalent:
\[ E = mc^2 \]
\[ = 0.001 \times (3 \times 10^8)^2 \]
\[ = 0.001 \times 9 \times 10^{16} \]
\[ = 9 \times 10^{13} \, \text{J} \]
For comparison, this is equivalent to:
\[ \frac{9 \times 10^{13}}{3.6 \times 10^6} \approx 25 \, \text{million kWh} \]

This demonstrates the enormous amount of energy contained in even a small amount of matter.

Relativistic Velocity Transformations

🚀 Velocity Addition Formula

In special relativity, velocities don't add simply as in classical physics. If an object moves with velocity \(u'\) in frame S', which itself moves with velocity \(v\) relative to frame S, then the velocity \(u\) of the object in frame S is:

\[ u = \frac{u' + v}{1 + \frac{u' v}{c^2}} \]

This formula ensures that the resulting velocity never exceeds the speed of light, regardless of how close \(u'\) and \(v\) are to \(c\).

🧮 Derivation of Velocity Transformation

Step 1: Lorentz Transformation
\[ dx' = \gamma(dx - v dt) \]
\[ dt' = \gamma\left(dt - \frac{v dx}{c^2}\right) \]
Step 2: Velocity Definition
\[ u' = \frac{dx'}{dt'} \]
\[ = \frac{\gamma(dx - v dt)}{\gamma\left(dt - \frac{v dx}{c^2}\right)} \]
Step 3: Simplify
\[ u' = \frac{dx - v dt}{dt - \frac{v dx}{c^2}} \]
\[ = \frac{\frac{dx}{dt} - v}{1 - \frac{v}{c^2} \frac{dx}{dt}} \]
Step 4: Final Formula
\[ u' = \frac{u - v}{1 - \frac{u v}{c^2}} \]

Solving for \(u\):

\[ u = \frac{u' + v}{1 + \frac{u' v}{c^2}} \]
Sample Problem: Relativistic Velocity Addition

A spaceship moves at 0.8c relative to Earth. If the spaceship launches a probe at 0.6c relative to itself in the same direction, what is the probe's speed relative to Earth?

Given:
\[ v = 0.8c \]
\[ u' = 0.6c \]
Using relativistic velocity addition:
\[ u = \frac{u' + v}{1 + \frac{u' v}{c^2}} \]
\[ = \frac{0.6c + 0.8c}{1 + \frac{(0.6c)(0.8c)}{c^2}} \]
\[ = \frac{1.4c}{1 + 0.48} \]
\[ = \frac{1.4c}{1.48} \]
\[ = 0.946c \]

Note that classical addition would give \(0.6c + 0.8c = 1.4c\), which exceeds the speed of light. The relativistic formula correctly gives a result less than \(c\).

Sample Problems and Applications

Problem 1: Muon Decay

Muons created in the upper atmosphere have a mean lifetime of 2.2 μs in their rest frame. If they travel at 0.999c, how far can they travel before decaying?

Given:
\[ \tau_0 = 2.2 \times 10^{-6} \, \text{s} \]
\[ v = 0.999c \]
Lorentz factor:
\[ \gamma = \frac{1}{\sqrt{1 - (0.999)^2}} \]
\[ = \frac{1}{\sqrt{1 - 0.998001}} \]
\[ = \frac{1}{\sqrt{0.001999}} \]
\[ = \frac{1}{0.04471} \]
\[ \approx 22.37 \]
Lifetime in Earth frame:
\[ \tau = \gamma \tau_0 \]
\[ = 22.37 \times 2.2 \times 10^{-6} \]
\[ = 4.92 \times 10^{-5} \, \text{s} \]
Distance traveled:
\[ d = v \tau \]
\[ = 0.999 \times 3 \times 10^8 \times 4.92 \times 10^{-5} \]
\[ = 1.475 \times 10^4 \, \text{m} \]
\[ = 14.75 \, \text{km} \]

Without time dilation, the distance would be only \(0.999c \times 2.2 \times 10^{-6} = 659 \, \text{m}\). The observed distance of several kilometers provides experimental confirmation of time dilation.

🚀 GPS Systems

GPS satellites must account for both special and general relativistic effects to maintain accuracy. Special relativistic time dilation causes satellite clocks to run slower by about 7 μs per day.

⚛️ Particle Accelerators

In particle accelerators like the LHC, particles reach speeds extremely close to c. Relativistic effects are essential for calculating their behavior, energy requirements, and collision outcomes.

🔭 Astrophysics

Relativistic effects explain phenomena like the precession of Mercury's orbit, gravitational lensing, and the behavior of matter near black holes where velocities approach c.

Frequently Asked Questions

❓ Why can't anything travel faster than light?

According to special relativity, as an object with mass approaches the speed of light:

  • Its relativistic mass increases toward infinity
  • The energy required for further acceleration approaches infinity
  • Time dilation means the object would experience infinite time contraction
  • Length contraction would reduce its length to zero in the direction of motion

These effects make it physically impossible for any object with mass to reach or exceed the speed of light. Only massless particles like photons can travel at exactly c.

❓ Does special relativity violate causality?

No, special relativity preserves causality (the principle that cause must precede effect) through several mechanisms:

  • The speed of light acts as an ultimate speed limit for information transfer
  • Events that are causally connected always maintain their time order in all reference frames
  • Only events that are spacelike separated (cannot influence each other) can have their time order reversed in different frames

This ensures that no observer can see an effect happen before its cause.

❓ How is special relativity different from general relativity?

Special and general relativity address different physical situations:

  • Special Relativity (1905): Deals with inertial frames (non-accelerating) in the absence of gravity
  • General Relativity (1915): Extends the principles to accelerated frames and incorporates gravity through the curvature of spacetime

Special relativity is a special case of general relativity that applies when gravitational effects are negligible. General relativity reduces to special relativity in local inertial frames.

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