Special Relativity

Introduction: The Crisis of Common Sense

For centuries, physics was built on the intuition of Isaac Newton. Space was a rigid stage, and time was a universal clock ticking the same for everyone, everywhere. This view is summarized by Galilean Relativity.

However, in the late 19th century, James Clerk Maxwell’s equations of electromagnetism revealed a problem: they predicted that the speed of light, $c$, is a fundamental constant ($c \approx 3 \times 10^8$ m/s). But if you run towards a light beam, shouldn’t you measure it moving slower relative to you? Maxwell said “no”, Newton said “yes”.

Albert Einstein solved this in 1905 with Special Relativity, fundamentally changing our understanding of reality.

Two Worlds: Galilean vs. Minkowski Space

To understand relativity, we must compare how we translate coordinates between two observers. Imagine two frames of reference: 1. Frame $S$ (e.g., a person on a platform). 2. Frame $S'$ (e.g., a person on a train) moving with velocity $v$ along the $x$-axis relative to $S$.

1. The Classical World (Galilean Transformation)

In classical mechanics, time is absolute ($t' = t$). If an event happens at $(x, t)$ in frame $S$, the observer in $S'$ sees it at:

\[ \begin{cases} x' = x - vt \\ y' = y \\ z' = z \\ t' = t \end{cases} \]

Velocity Addition: If a ball is thrown with speed $u_x'$ inside the train ($S'$), its speed seen from the platform ($S$) is simply: \[u_x = u_x' + v\]

If the ball is a photon ($u_x' = c$), then $u_x = c + v$. This contradicts experiment and Maxwell’s equations!

2. The Relativistic World (Lorentz Transformation)

Einstein postulated that the speed of light $c$ is constant in all inertial frames. For this to be true, time and space must mix. The correct transformation is the Lorentz Transformation:

First, we define two helper variables: \[ \beta = \frac{v}{c} \quad \text{(Speed parameter)} \] \[ \gamma = \frac{1}{\sqrt{1 - \beta^2}} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \quad \text{(Lorentz factor)} \] Note: As $v \to 0$, $\gamma \to 1$, and we recover Galilean mechanics. As $v \to c$, $\gamma \to \infty$.

The transformation equations are:

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

Key Difference: Look at the equation for $t'$. Time is no longer absolute! Your time $t'$ depends on my position $x$. This is the “loss of simultaneity”.

Spacetime Interval (The Invariant)

In 3D Euclidean space (Galilean), the distance between two points is invariant upon rotation: \[\Delta r^2 = \Delta x^2 + \Delta y^2 + \Delta z^2\]

In Minkowski Space (Relativity), space and time are fused into 4D Spacetime. The invariant “distance” is called the Spacetime Interval ($s^2$):

\[ \Delta s^2 = (c\Delta t)^2 - (\Delta x^2 + \Delta y^2 + \Delta z^2) \]

All observers, regardless of their speed, agree on the value of $\Delta s^2$. - If $\Delta s^2 > 0$: Timelike separation (causally connected events). - If $\Delta s^2 < 0$: Spacelike separation (no causal connection). - If $\Delta s^2 = 0$: Lightlike separation (connected by a light ray).

Consequences of Lorentz Transformations

Using the Lorentz equations, we can derive the strange effects of high-speed travel.

1. Time Dilation

Scenario: An observer in frame $S'$ (the moving train) has a clock at a fixed position $x'$. They measure the time interval between two ticks (events) as the Proper Time $\Delta t_0$. * Event 1: $(x', t'_1)$ * Event 2: $(x', t'_2)$ * Duration: $\Delta t' = t'_2 - t'_1 = \Delta t_0$

What does the observer in $S$ (platform) measure? We use the inverse Lorentz transformation for time ($t = \gamma(t' + \frac{v}{c^2}x')$):

\[ t_1 = \gamma (t'_1 + \frac{v}{c^2}x') \] \[ t_2 = \gamma (t'_2 + \frac{v}{c^2}x') \]

Subtracting these gives:

\[ \Delta t = t_2 - t_1 = \gamma (t'_2 - t'_1) \]

\[ \Delta t = \gamma \Delta t_0 \]

Conclusion: Since $\gamma \geq 1$, $\Delta t \geq \Delta t_0$. Moving clocks run slower. The observer on the platform sees the train’s clock ticking slowly. The “proper time” (measured by the clock itself) is always the shortest time interval.

2. Length Contraction

Scenario: A stick is at rest in frame $S'$ (the moving train). Its length is measured in $S'$ as the Proper Length $L_0$. * End 1: $x'_1$ * End 2: $x'_2$ * Length: $L_0 = x'_2 - x'_1$

The observer in $S$ (platform) wants to measure the length. To measure the length of a moving object, you must record the positions of both ends simultaneously ($\Delta t = 0$ in frame $S$). We use the Lorentz transformation for $x'$: $x' = \gamma(x - vt)$.

\[ x'_1 = \gamma (x_1 - v t_1) \] \[ x'_2 = \gamma (x_2 - v t_2) \]

Since the measurement in $S$ is simultaneous, $t_1 = t_2$. Subtracting the equations:

\[ x'_2 - x'_1 = \gamma (x_2 - x_1) \]

\[ L_0 = \gamma L \]

Rearranging for $L$:

\[ L = \frac{L_0}{\gamma} \]

Conclusion: Since $\gamma \geq 1$, $L \leq L_0$. Moving objects appear shorter along the direction of motion.

Concrete Example: Cosmic Ray Muons

This is not just theory; we observe it daily. Muons are unstable particles created in the upper atmosphere (approx. 10 km up) by cosmic rays. * Muon lifetime (at rest): $\tau \approx 2.2 \mu s$. * Muon speed: $v \approx 0.998c$ ($\gamma \approx 15$).

Classical Prediction: Distance traveled $= v \times \tau \approx (3 \times 10^8 \text{ m/s}) \times (2.2 \times 10^{-6} \text{ s}) \approx 660 \text{ meters}$. Muons should decay long before reaching the ground. Yet, we detect them at sea level! Why?

Explanation 1: From Earth’s Perspective (Time Dilation)

To us on Earth, the muon is a moving clock. Its internal “watch” runs slower by a factor of $\gamma$. * Dilated lifetime: $\Delta t = \gamma \tau = 15 \times 2.2 \mu s = 33 \mu s$. * Distance traveled: $d = v \times \Delta t \approx c \times 33 \mu s \approx 9900 \text{ meters}$. Result: The muon lives long enough to reach the ground.

Explanation 2: From Muon’s Perspective (Length Contraction)

To the muon, it is at rest, and the Earth is rushing towards it at $0.998c$. The distance to the ground ($L_0 = 10 \text{ km}$) is length-contracted. * Contracted distance: $L = \frac{L_0}{\gamma} = \frac{10000 \text{ m}}{15} \approx 666 \text{ meters}$. * Time to hit Earth: $t = \frac{L}{v} \approx \frac{666}{c} \approx 2.22 \mu s$. Result: The ground reaches the muon just before it decays.

Both observers agree on the outcome (the muon hits the ground), but they disagree on why (distorted time vs. distorted space).

Relativistic Velocity Addition

If you run at $0.8c$ inside a rocket moving at $0.8c$, you do not move at $1.6c$. The Galilean formula $u = u' + v$ fails. The correct formula derived from Lorentz transformations is:

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

If we substitute $u' = c$ (a photon fired from the rocket):

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

The speed of light is $c$ for everyone, preserving the fundamental postulate.

Mass-Energy Equivalence

Finally, relativity modifies Newton’s Second Law. As an object approaches $c$, it becomes harder to accelerate. This leads to the most famous equation in physics. Total energy $E$ is related to momentum $p$ and rest mass $m_0$:

\[ E^2 = (pc)^2 + (m_0 c^2)^2 \]

For a particle at rest ($p=0$): \[ E = m_0 c^2 \]

This implies that mass is simply a concentrated form of energy.

Summary Table

Concept	Galilean Relativity (Classical)	Special Relativity (Einstein)
Time	Absolute ($t' = t$)	Relative ($t' \neq t$)
Space	Absolute ($L' = L$)	Relative ($L = L_0 / \gamma$)
Speed of Light	Variable ($c \pm v$)	Constant ($c$)
Transformation	$x' = x - vt$	$x' = \gamma(x - vt)$
Invariant	Distance $\Delta r^2$	Interval $c^2\Delta t^2 - \Delta r^2$

--- title: Special Relativity format: html: theme: flatly toc: true toc-depth: 3 highlight-style: tango code-line-numbers: true code-fold: true code-summary: "Show the code" code-tools: true code-block-bg: "rgba(42, 174, 42, 0.02)" code-block-border-left: "#2aae2a" code-language-label: true css: styles.css math: mathjax self-contained: true other-links: - text: Main page href: https://dchorazkiewicz.github.io/Mathematics_Physics_Lectures --- ## Introduction: The Crisis of Common Sense For centuries, physics was built on the intuition of Isaac Newton. Space was a rigid stage, and time was a universal clock ticking the same for everyone, everywhere. This view is summarized by **Galilean Relativity**. However, in the late 19th century, James Clerk Maxwell's equations of electromagnetism revealed a problem: they predicted that the speed of light, $c$, is a fundamental constant ($c \approx 3 \times 10^8$ m/s). But if you run towards a light beam, shouldn't you measure it moving slower relative to you? Maxwell said "no", Newton said "yes". Albert Einstein solved this in 1905 with **Special Relativity**, fundamentally changing our understanding of reality. ## Two Worlds: Galilean vs. Minkowski Space To understand relativity, we must compare how we translate coordinates between two observers. Imagine two frames of reference: 1. **Frame $S$** (e.g., a person on a platform). 2. **Frame $S'$** (e.g., a person on a train) moving with velocity $v$ along the $x$-axis relative to $S$. ### 1. The Classical World (Galilean Transformation) In classical mechanics, time is absolute ($t' = t$). If an event happens at $(x, t)$ in frame $S$, the observer in $S'$ sees it at: $$ \begin{cases} x' = x - vt \\ y' = y \\ z' = z \\ t' = t \end{cases} $$ **Velocity Addition:** If a ball is thrown with speed $u_x'$ inside the train ($S'$), its speed seen from the platform ($S$) is simply: $$u_x = u_x' + v$$ If the ball is a photon ($u_x' = c$), then $u_x = c + v$. This contradicts experiment and Maxwell's equations! ### 2. The Relativistic World (Lorentz Transformation) Einstein postulated that **the speed of light $c$ is constant in all inertial frames**. For this to be true, time and space *must* mix. The correct transformation is the **Lorentz Transformation**: First, we define two helper variables: $$ \beta = \frac{v}{c} \quad \text{(Speed parameter)} $$ $$ \gamma = \frac{1}{\sqrt{1 - \beta^2}} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \quad \text{(Lorentz factor)} $$ *Note: As $v \to 0$, $\gamma \to 1$, and we recover Galilean mechanics. As $v \to c$, $\gamma \to \infty$.* The transformation equations are: $$ \begin{cases} x' = \gamma (x - vt) \\ y' = y \\ z' = z \\ t' = \gamma (t - \frac{v}{c^2}x) \end{cases} $$ **Key Difference:** Look at the equation for $t'$. Time is no longer absolute! Your time $t'$ depends on my position $x$. This is the "loss of simultaneity". ### Spacetime Interval (The Invariant) In 3D Euclidean space (Galilean), the distance between two points is invariant upon rotation: $$\Delta r^2 = \Delta x^2 + \Delta y^2 + \Delta z^2$$ In **Minkowski Space** (Relativity), space and time are fused into 4D Spacetime. The invariant "distance" is called the **Spacetime Interval** ($s^2$): $$ \Delta s^2 = (c\Delta t)^2 - (\Delta x^2 + \Delta y^2 + \Delta z^2) $$ All observers, regardless of their speed, agree on the value of $\Delta s^2$. - If $\Delta s^2 > 0$: **Timelike** separation (causally connected events). - If $\Delta s^2 < 0$: **Spacelike** separation (no causal connection). - If $\Delta s^2 = 0$: **Lightlike** separation (connected by a light ray). ## Consequences of Lorentz Transformations Using the Lorentz equations, we can derive the strange effects of high-speed travel. ### 1. Time Dilation **Scenario:** An observer in frame $S'$ (the moving train) has a clock at a fixed position $x'$. They measure the time interval between two ticks (events) as the **Proper Time** $\Delta t_0$. * Event 1: $(x', t'_1)$ * Event 2: $(x', t'_2)$ * Duration: $\Delta t' = t'_2 - t'_1 = \Delta t_0$ What does the observer in $S$ (platform) measure? We use the inverse Lorentz transformation for time ($t = \gamma(t' + \frac{v}{c^2}x')$): $$ t_1 = \gamma (t'_1 + \frac{v}{c^2}x') $$ $$ t_2 = \gamma (t'_2 + \frac{v}{c^2}x') $$ Subtracting these gives: $$ \Delta t = t_2 - t_1 = \gamma (t'_2 - t'_1) $$ $$ \Delta t = \gamma \Delta t_0 $$ **Conclusion:** Since $\gamma \geq 1$, $\Delta t \geq \Delta t_0$. **Moving clocks run slower.** The observer on the platform sees the train's clock ticking slowly. The "proper time" (measured by the clock itself) is always the shortest time interval. ### 2. Length Contraction **Scenario:** A stick is at rest in frame $S'$ (the moving train). Its length is measured in $S'$ as the **Proper Length** $L_0$. * End 1: $x'_1$ * End 2: $x'_2$ * Length: $L_0 = x'_2 - x'_1$ The observer in $S$ (platform) wants to measure the length. To measure the length of a moving object, you must record the positions of both ends **simultaneously** ($\Delta t = 0$ in frame $S$). We use the Lorentz transformation for $x'$: $x' = \gamma(x - vt)$. $$ x'_1 = \gamma (x_1 - v t_1) $$ $$ x'_2 = \gamma (x_2 - v t_2) $$ Since the measurement in $S$ is simultaneous, $t_1 = t_2$. Subtracting the equations: $$ x'_2 - x'_1 = \gamma (x_2 - x_1) $$ $$ L_0 = \gamma L $$ Rearranging for $L$: $$ L = \frac{L_0}{\gamma} $$ **Conclusion:** Since $\gamma \geq 1$, $L \leq L_0$. **Moving objects appear shorter** along the direction of motion. ## Concrete Example: Cosmic Ray Muons This is not just theory; we observe it daily. Muons are unstable particles created in the upper atmosphere (approx. 10 km up) by cosmic rays. * Muon lifetime (at rest): $\tau \approx 2.2 \mu s$. * Muon speed: $v \approx 0.998c$ ($\gamma \approx 15$). **Classical Prediction:** Distance traveled $= v \times \tau \approx (3 \times 10^8 \text{ m/s}) \times (2.2 \times 10^{-6} \text{ s}) \approx 660 \text{ meters}$. Muons should decay long before reaching the ground. Yet, we detect them at sea level! Why? ### Explanation 1: From Earth's Perspective (Time Dilation) To us on Earth, the muon is a moving clock. Its internal "watch" runs slower by a factor of $\gamma$. * Dilated lifetime: $\Delta t = \gamma \tau = 15 \times 2.2 \mu s = 33 \mu s$. * Distance traveled: $d = v \times \Delta t \approx c \times 33 \mu s \approx 9900 \text{ meters}$. **Result:** The muon lives long enough to reach the ground. ### Explanation 2: From Muon's Perspective (Length Contraction) To the muon, it is at rest, and the Earth is rushing towards it at $0.998c$. The distance to the ground ($L_0 = 10 \text{ km}$) is length-contracted. * Contracted distance: $L = \frac{L_0}{\gamma} = \frac{10000 \text{ m}}{15} \approx 666 \text{ meters}$. * Time to hit Earth: $t = \frac{L}{v} \approx \frac{666}{c} \approx 2.22 \mu s$. **Result:** The ground reaches the muon just before it decays. Both observers agree on the outcome (the muon hits the ground), but they disagree on *why* (distorted time vs. distorted space). ## Relativistic Velocity Addition If you run at $0.8c$ inside a rocket moving at $0.8c$, you do not move at $1.6c$. The Galilean formula $u = u' + v$ fails. The correct formula derived from Lorentz transformations is: $$ u = \frac{u' + v}{1 + \frac{u' v}{c^2}} $$ If we substitute $u' = c$ (a photon fired from the rocket): $$ u = \frac{c + v}{1 + \frac{c v}{c^2}} = \frac{c + v}{1 + \frac{v}{c}} = \frac{c + v}{\frac{c + v}{c}} = c $$ The speed of light is $c$ for everyone, preserving the fundamental postulate. ## Mass-Energy Equivalence Finally, relativity modifies Newton's Second Law. As an object approaches $c$, it becomes harder to accelerate. This leads to the most famous equation in physics. Total energy $E$ is related to momentum $p$ and rest mass $m_0$: $$ E^2 = (pc)^2 + (m_0 c^2)^2 $$ For a particle at rest ($p=0$): $$ E = m_0 c^2 $$ This implies that mass is simply a concentrated form of energy. ## Summary Table | Concept | Galilean Relativity (Classical) | Special Relativity (Einstein) | | :--- | :--- | :--- | | **Time** | Absolute ($t' = t$) | Relative ($t' \neq t$) | | **Space** | Absolute ($L' = L$) | Relative ($L = L_0 / \gamma$) | | **Speed of Light** | Variable ($c \pm v$) | Constant ($c$) | | **Transformation** | $x' = x - vt$ | $x' = \gamma(x - vt)$ | | **Invariant** | Distance $\Delta r^2$ | Interval $c^2\Delta t^2 - \Delta r^2$ |

Concept	Galilean Relativity (Classical)	Special Relativity (Einstein)
Time	Absolute (\(t' = t\))	Relative (\(t' \neq t\))
Space	Absolute (\(L' = L\))	Relative (\(L = L_0 / \gamma\))
Speed of Light	Variable (\(c \pm v\))	Constant (\(c\))
Transformation	\(x' = x - vt\)	\(x' = \gamma(x - vt)\)
Invariant	Distance \(\Delta r^2\)	Interval \(c^2\Delta t^2 - \Delta r^2\)