Physics | Galilean and special relativity (HL)

Galilean and Special Relativity (HL) - Space, Time, and Motion

Hello future Einstein! This chapter is where classical physics gives way to the truly mind-bending concepts that govern the high-speed universe. Don't worry if this seems tricky at first—you're challenging centuries of classical thought! We will start with common sense (Galilean relativity) and then shatter it with the revolutionary ideas of Albert Einstein (Special Relativity). This is a Higher Level topic, meaning we will dive deep into the mathematics that describes how space and time literally warp when you travel fast enough.

Prerequisite Check: Before starting, make sure you are comfortable with basic kinematics (displacement, velocity) and the concept of frames of reference.

1. Galilean Relativity: The Classical View

Galilean relativity is your everyday intuition about motion. It worked perfectly for centuries until we tried to apply it to light.

1.1 Inertial Frames of Reference

An inertial frame of reference is a frame (coordinate system) that is either at rest or moving with a constant velocity. Crucially, in an inertial frame, Newton's first law (the law of inertia) holds true: objects not subjected to a net force remain at rest or in uniform motion.

• Example: If you are standing still on the ground, you are in an inertial frame. If you are on a train moving at a steady 100 km/h in a straight line, you are also in an inertial frame.
• Non-Example: A spinning carousel or a braking car are non-inertial frames because objects inside them experience apparent forces (like centrifugal force or the force that throws you forward when braking).

1.2 The Principle of Galilean Relativity

The core principle is simple: The laws of mechanics (physics involving forces and motion) are the same in all inertial frames of reference.

Analogy: Imagine you are below deck on a cruise ship moving smoothly. You cannot tell if the ship is moving or stationary just by doing simple experiments (like throwing a ball or dropping a pencil). The physics is the same in both frames.

1.3 Galilean Transformations (Classical Velocity Addition)

This tells us how velocities transform between two inertial frames.

Suppose Frame S' is moving at a velocity \(v\) relative to Frame S (the stationary ground frame). If an object moves at velocity \(u'\) in Frame S', its velocity \(u\) in Frame S is:

$$ u = u' + v $$

Example: A train (Frame S') moves at \(v = 20 \text{ m/s}\). A passenger throws a ball forward at \(u' = 5 \text{ m/s}\) inside the train. Someone standing on the ground (Frame S) sees the ball moving at \(u = 5 + 20 = 25 \text{ m/s}\). Simple addition!

2. The Problem: The Speed of Light

In the late 19th century, physicists expected light (an electromagnetic wave) to obey Galilean transformations. If a light source moves toward you, you should measure its speed as \(c + v\). But experiments disagreed.

2.1 The Michelson-Morley Experiment (The Null Result)

Scientists believed light propagated through a substance called the luminiferous aether. The Earth's motion through this aether should cause the measured speed of light to change depending on the direction of measurement.

The famous Michelson-Morley experiment (1887) tried to measure this difference.

The Result: No matter how the Earth was moving or what direction the light traveled, the speed of light was measured to be exactly the same constant value, \(c\).

This was a disaster for classical physics. If the speed of light doesn't change, then Galilean velocity addition must be wrong!

3. Einstein's Special Relativity (SR)

In 1905, Albert Einstein published his theory of Special Relativity, resolving the crisis by proposing two postulates that fundamentally redefine our concepts of space and time.

3.1 Postulate 1: The Principle of Relativity

The laws of physics are the same for all observers in all inertial frames of reference. (This is the same as Galilean's first postulate, but Einstein extended it to *all* laws of physics, including electromagnetism/light, not just mechanics.)

Meaning: You cannot perform any physical experiment (mechanical, electrical, optical) to determine your absolute state of motion.

3.2 Postulate 2: The Constancy of the Speed of Light

The speed of light in a vacuum, \(c\), is the same for all inertial observers, regardless of the motion of the source or the observer.

$$ c \approx 3.00 \times 10^8 \text{ m/s} $$

This is the radical idea! If a spacecraft is traveling at \(0.5c\) and turns on its headlights, an observer standing still will measure the speed of that light as \(c\), not \(1.5c\). The speed of light is the universal speed limit and is constant.

4. Consequences of Special Relativity

If the speed of light must be constant, then space and time itself must change to make that happen. These changes are only noticeable at relativistic speeds (speeds approaching \(c\)).

4.1 The Lorentz Factor (\(\gamma\))

All relativistic effects are governed by the Lorentz factor (\(\gamma\)). You need to be very familiar with this equation:

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

• Here, \(v\) is the relative speed between the two frames.
• If \(v\) is small (like a car or plane), \(v^2/c^2\) is nearly zero, so \(\gamma\) is almost exactly 1. Classical physics holds true!
• If \(v\) approaches \(c\), the denominator approaches zero, and \(\gamma\) approaches infinity.

4.2 Time Dilation

Time Dilation is the effect where a moving clock runs slower than a stationary clock.

• Proper Time (\(\Delta t_0\)): This is the time interval measured by an observer at rest relative to the event (the "clock" is in the same frame as the observer). This is always the shortest measured time.
• Dilated Time (\(\Delta t\)): This is the time interval measured by an observer moving relative to the event.

$$ \Delta t = \gamma \Delta t_0 $$

Since \(\gamma\) is always greater than or equal to 1, \(\Delta t\) is always greater than \(\Delta t_0\). Time has been stretched (dilated) for the moving observer.

Real-World Example: Subatomic particles called muons are created high in the atmosphere. They live for a very short time (\(\Delta t_0\)). However, due to time dilation caused by their high speed, we measure them lasting much longer (\(\Delta t\)), allowing them to reach the Earth's surface—a proof that relativity is real!

4.3 Length Contraction

Length Contraction is the effect where the length of an object appears shorter when measured by an observer moving relative to it.

• Proper Length (\(L_0\)): This is the length measured by an observer at rest relative to the object. This is always the longest measured length.
• Contracted Length (\(L\)): This is the length measured by an observer moving relative to the object.

$$ L = \frac{L_0}{\gamma} $$

Since \(\gamma\) is always greater than or equal to 1, \(L\) is always less than \(L_0\). The length contracts only in the direction of motion.

Analogy: If a spaceship flies past Earth at 0.9c, observers on Earth would see the ship as squashed (shorter) along its direction of travel, while the astronauts inside would measure their ship's normal length \(L_0\).

5. Lorentz Transformations (HL Mandatory Content)

The Galilean transformations assumed that coordinates (x, y, z) and time (t) were the same in both frames. This is false in Special Relativity. The Lorentz Transformations are the correct set of equations that relate the coordinates of an event in one inertial frame (S) to another frame (S').

Assume Frame S' moves along the positive x-axis of Frame S with velocity \(v\).

If an event occurs at \((x, t)\) in Frame S, its coordinates \((x', t')\) in Frame S' are:

5.1 Transformation of Position (x)

$$ x' = \gamma (x - vt) $$

Note: If \(v \ll c\), then \(\gamma \approx 1\), and this simplifies back to the classical Galilean transformation \(x' = x - vt\).

5.2 Transformation of Time (t)

This is the radical part—time coordinates are mixed with position coordinates!

$$ t' = \gamma \left(t - \frac{vx}{c^2}\right) $$

Meaning: Two events that happen simultaneously (\(\Delta t = 0\)) in Frame S will generally *not* be simultaneous in Frame S', unless they happen at the same location (\(\Delta x = 0\)). This is called Relativity of Simultaneity.

5.3 Relativistic Velocity Addition (HL)

Since the speed of light must remain \(c\) for all observers, we cannot simply add velocities classically. We must use the relativistic velocity addition formula.

If Frame S' moves at velocity \(v\) relative to S, and an object moves at \(u'\) in S', its velocity \(u\) measured in S is:

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

Common Mistake to Avoid: Make sure you use the denominator \(1 + (u'v)/c^2\). This term ensures that if either \(u'\) or \(v\) is \(c\), the resulting velocity \(u\) is also exactly \(c\). For example, if \(u' = c\):

$$ u = \frac{c + v}{1 + \frac{cv}{c^2}} = \frac{c + v}{1 + \frac{v}{c}} = \frac{c(1 + v/c)}{(1 + v/c)} = c $$

5.4 Relativistic Momentum (HL)

Classical momentum is \(p = mv\). For momentum to be conserved in all inertial frames, we must introduce the Lorentz factor into the momentum definition.

The relativistic momentum (\(p\)) is defined as:

$$ p = \gamma m_0 v $$

Here, \(m_0\) is the rest mass (mass measured when the object is stationary). As velocity \(v\) increases, \(\gamma\) increases, meaning the momentum increases much faster than predicted by classical physics. This explains why accelerating an object to \(c\) requires infinite momentum (and therefore infinite energy).

6. Mass-Energy Equivalence

One of the most profound outcomes of Special Relativity is the relationship between mass and energy.

The total energy \(E\) of an object is related to its rest mass \(m_0\) and its speed \(v\):

$$ E = \gamma m_0 c^2 $$

6.1 Rest Energy and \(E=mc^2\)

If the object is at rest (\(v = 0\)), then \(\gamma = 1\), and the equation simplifies to the world's most famous formula:

$$ E_0 = m_0 c^2 $$

This equation states that even a particle at rest possesses a huge amount of energy—its rest energy (\(E_0\)). Mass and energy are interchangeable. When mass disappears (like in nuclear fusion or fission), a corresponding amount of energy is released.

6.2 Relativistic Kinetic Energy

The total energy of a moving particle is \(E = E_0 + E_k\). Therefore, the relativistic kinetic energy \(E_k\) is:

$$ E_k = E - E_0 = \gamma m_0 c^2 - m_0 c^2 $$

$$ E_k = m_0 c^2 (\gamma - 1) $$

For small velocities, this formula approaches the classical definition \(E_k = \frac{1}{2} m_0 v^2\). But at high speeds, the energy required to accelerate the object increases dramatically due to the rapidly growing \(\gamma\).

Did you know? The theory of Special Relativity forms the basis for how particle accelerators work. Engineers must use relativistic formulas to calculate the correct momentum and energy needed to control particles moving close to the speed of light.