DP IB Physics: SL

A. Space, time and motion

A.5 Galilean and special relativity

DP IB Physics: SL A. Space, time and motion A.5 Galilean and special relativity Understandings Students should understand:
a)	Reference frames
b)	That Newton’s laws of motion are the same in all inertial reference frames and this is known as Galilean relativity
c)	That in Galilean relativity the position [math]x'[/math] and time [math]t'[/math] of an event are given by [math]x’ = x – vt[/math] and [math]t’ = t[/math]
d)	That Galilean transformation equations lead to the velocity addition equation as given by [math]u’ = u – v[/math]
e)	The two postulates of special relativity
f)	That the postulates of special relativity lead to the Lorentz transformation equations for the coordinates of an event in two inertial reference frames as given by [math]x’ = \gamma (x – vt) \\ t’ = \gamma \left(t – \frac{vx}{c^2}\right)[/math] Where [math]\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}[/math]
g)	That Lorentz transformation equations lead to the relativistic velocity addition equation as given by [math]u’ = \frac{u – v}{1 – \frac{uv}{c^2}}[/math]
h)	That the space–time interval between two events is an invariant quantity as given by [math](∆s)^2 = (c∆t)^2 – (∆x)^2[/math]
i)	Proper time interval and proper length
j)	Time dilation as given by [math]Δt = γΔt_0[/math]
k)	Length contraction as given by [math]L = \frac{L_0}{γ}[/math]
l)	The relativity of simultaneity
m)	Space–time diagrams
n)	That the angle between the world line of a moving particle and the time axis on a space–time diagram is related to the particle’s speed as given by [math]tan θ = \frac{v}{c}[/math]
o)	That muon decay experiments provide experimental evidence for time dilation and length contraction.

(a) Reference frames:
A reference frame is a coordinate system used to measure the position, velocity, and acceleration of objects. There are two main types:
1. Inertial Reference Frame
– A frame in which Newton’s laws hold true.
– It moves at a constant velocity (no acceleration) relative to another inertial frame.
– Example: A train moving at constant speed or a stationary observer on the ground.
2. Non-Inertial Reference Frame
– A frame that is accelerating or rotating.
– Newton’s laws appear to be violated unless fictitious (pseudo) forces are introduced.
– Example: A car taking a sharp turn (passengers feel a force pushing them sideways).
Figure 1 Reference frames
(b) Newton’s Laws in Inertial Reference Frames
Newton’s laws of motion apply equally in all inertial reference frames.
First Law (Law of Inertia):
– Objects at rest stay at rest, and objects in motion continue in motion unless acted upon by an external force.
– Holds true in all inertial frames.
Figure 2 Law of Inertia
Second Law (Force and Acceleration):
[math]F = ma[/math]
applies in all inertial frames.
– The acceleration of an object is the same in all inertial frames.
Third Law (Action-Reaction):
– For every action, there is an equal and opposite reaction.
– Thus, in all inertial frames, the laws of physics remain unchanged.
Figure 3 Action and reaction
(c) Galilean Relativity
Galilean relativity states that:
– The laws of motion are the same in all inertial frames.
– There is no absolute reference frame; only relative motion
– If an observer moves at a constant velocity, they experience the same physics as one at rest.
Figure 4 Galilean relativity
⇒ Example:
A person on a moving train throws a ball straight up.
– To them, it goes up and down.
– To an observer outside, the ball follows a curved path (parabolic motion).
– But both describe the motion using Newton’s laws!
This confirms that all inertial observers see the same physics.
⇒ Galilean Transformations
Galilean transformations allow us to convert measurements from one inertial frame to another moving at a constant velocity v.
Let’s define two reference frames:
S (stationary observer)
S’ (moving observer with velocity v)
The Galilean transformation equations are:
1. Position transformation:
[math]x’ = x – vt[/math]
Where:
– x′ = position in moving frame
– x = position in stationary frame
– v = velocity of moving frame relative to stationary frame
– t = time (same in both frames)
2. Time transformation:
[math]t′ = t[/math]
– Time is absolute in Galilean relativity (unlike Einstein’s relativity).
3. Velocity transformation:
[math]v’ = v – V[/math]
Where:
– v′ = velocity in moving frame
– v = velocity in stationary frame
– V = velocity of moving frame
Example:
A car moves at 20 m/s. A truck moves at 30 m/s in the same direction.
To the stationary observer, the truck moves at 30 m/s.
To the car driver, the truck moves at 30 – 20 = 10 m/s.
Galilean transformations allow us to switch between reference frames easily.
⇒ Limitations of Galilean Relativity
Galilean relativity works well at low speeds but fails at high speeds (close to the speed of light).
– It assumes time is absolute, but in reality, time is relative (as shown in Einstein’s Special Relativity).
– It does not account for length contraction or time dilation.
For speeds near light speed (c), we must use Lorentz transformations, not Galilean transformations.
(d) Galilean transformation, velocity addition, and special relativity
⇒ Galilean Velocity Addition
The Galilean transformation equations describe how positions, velocities, and times change between two inertial reference frames moving at a constant velocity relative to each other.
If we have:
– A stationary observer (S)
– A moving observer (S’) traveling at velocity v relative to S
– An object moving at velocity u in S
Then, the velocity u′ of the object in the moving frame S′ is given by:
[math]u’ = u – v[/math]
⇒ Example:
Imagine a car traveling at 50 m/s relative to the ground. A person inside throws a ball forward at 10 m/s.
– To the car driver, the ball moves at 10 m/s.
– To a person standing outside, the ball moves at:
[math]u’ = 50 + 10 \\ u’ = 60 m/s[/math]
This shows how velocities add in classical mechanics. However, this rule fails at speeds close to the speed of light (c).
(e) The Postulates of Special Relativity
Albert Einstein formulated Special Relativity to explain inconsistencies in classical mechanics and electromagnetism. It is based on two fundamental postulates:
Figure 5 Special theory of Relativity
⇒ Postulate 1: The Principle of Relativity
The laws of physics are the same in all inertial reference frames.
– This extends Galilean Relativity but also includes Maxwell’s equations (electromagnetism).
– No experiment can distinguish whether a system is moving or at rest without referring to another system.
⇒ Example:
On a smooth-moving train, if you drop a ball, it falls straight down (same as if you were stationary). This means motion at a constant velocity is undetectable without looking outside.
⇒ Postulate 2: The Constancy of the Speed of Light
The speed of light in a vacuum (c) is the same for all observers, regardless of their motion or the motion of the source of light.
– This is a fundamental difference from Galilean relativity.
– Unlike normal velocities (which add up), light’s speed remains constant even if the observer is moving.
⇒ Example:
If a spaceship moves at 0.9c and turns on its headlights:
– Classical physics would predict the light moves at 9c.
– But Special Relativity states that the speed of light remains exactly ccc, not faster than ccc!
⇒ Galilean Transformations Fail at High Speeds
Using Galilean transformations:
[math]u’ = u – v[/math]
If u = c (speed of light), then:
[math]u’ = c – v[/math]
This suggests light should slow down if the observer moves toward it, but experiments show it doesn’t!
Special Relativity fixes this using the Lorentz Transformation:
[math]u’ = \frac{u – v}{1 – \frac{uv}{c^2}}[/math]
This ensures that if u = c, then u′ = c always.
⇒ Consequences of Special Relativity
1. Time Dilation:
– Moving clocks run slower than stationary ones.
– Given by:
[math]t’ = \frac{t}{\sqrt{1 – \frac{v^2}{c^2}}}[/math]
2. Length Contraction:
– Moving objects appear shorter in the direction of motion.
– Given by:
[math]L’ = L \sqrt{1 – \frac{v^2}{c^2}}[/math]
3. Relativistic Mass Increase:
– Mass increases as speed approaches c.
[math]m’ = \frac{m}{\sqrt{1 – \frac{v^2}{c^2}}}[/math]
4. Mass-Energy Equivalence:
– Energy and mass are related by:
[math]E = mc^2[/math]
(f) Lorentz transformations and relativistic velocity addition
⇒ The Need for Lorentz Transformations
In classical mechanics, the Galilean transformation equations describe how space and time coordinates change between two reference frames moving at a constant velocity relative to each other:
[math]x’ = x – vt \\ t’ = t[/math]
However, these equations assume that time is absolute (the same for all observers), which contradicts Einstein’s Special Relativity.
Figure 6 Lorentz transformations and relativistic velocity addition
Special Relativity states that:
– The laws of physics are the same in all inertial reference frames.
– The speed of light (c) is constant in all reference frames, regardless of the observer’s motion.
If the speed of light must remain constant, time and space must transform differently when moving at high velocities. This leads to the Lorentz Transformations.
(g) Lorentz Transformation Equations
The Lorentz transformation equations relate the space and time coordinates of an event as observed in two different inertial frames.
If an observer in frame S′ moves at velocity v relative to frame S along the x-axis, the transformations are:
[math]x’ = \gamma (x – vt) \\
t’ = \gamma \left(t – \frac{vx}{c^2}\right)[/math]
Where the Lorentz factor γ is:
[math]\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}[/math]
⇒ Explanation of Lorentz Transformations
Length Contraction
Objects moving relative to an observer appear shorter along the direction of motion.
The contracted length L′ is given by:
[math]L’ = L \sqrt{1 – \frac{v^2}{c^2}}[/math]
Time Dilation
A moving clock runs slower than a stationary one.
The dilated time t′ is:
[math]t’ = \frac{t}{\sqrt{1 – \frac{v^2}{c^2}}}[/math]
Relativity of Simultaneity
Events that appear simultaneous in one frame may not be simultaneous in another.
⇒ Derivation of the Relativistic Velocity Addition Formula
Galilean Velocity Addition (Classical Mechanics)
In Newtonian mechanics, if an object moves at velocity u in frame S, then in a frame S′ moving at velocity v relative to S, the velocity is:
[math]u’ = u – v[/math]
However, this fails for high speeds because it does not preserve the constancy of c.
Relativistic Velocity Addition
Using Lorentz transformations, the correct formula for velocity transformation is:
[math]u’ = \frac{u – v}{1 – \frac{uv}{c^2}}[/math]
⇒ Consequences of the Relativistic Velocity Addition
Speed of Light is Always c
If u = c, then:
[math]u’ = \frac{u – v}{1 – \frac{uv}{c^2}} \\
u’ = c[/math]
This ensures that light’s speed remains constant in all reference frames.
Velocities Add Differently at High Speeds
If u=8c and v=0.8c, the classical sum would be 1.6c (which is impossible).
Using relativistic velocity addition:
[math]u’ = \frac{u – v}{1 – \frac{uv}{c^2}} \\
u’ = \frac{0.8c – 0.8c}{1 – \frac{(0.8c)(0.8c)}{c^2}} \\
u’ = \frac{1.6c}{1 + 0.64} \\
u’ = \frac{1.6c}{1.64} \\
u’ \approx 0.976c[/math]
The result is still less than c.
(h) The Concept of Space-Time Interval
In special relativity, space and time are intertwined into a four-dimensional space-time continuum. Unlike Newtonian physics, where time is absolute, relativity shows that different observers may measure different distances and time intervals depending on their motion.
However, one quantity remains invariant for all observers: the space-time interval Δx between two events. This is defined as:
[math](\Delta s)^2 = (c \Delta t)^2 – (\Delta x)^2[/math]
– This equation is similar to the Pythagorean theorem, but with a minus sign due to the nature of space-time.
Figure 7 Space time interval
Properties of the Space-Time Interval
It remains the same for all observers in any inertial reference frame.
It determines whether events are time like, spacelike, or light like:
– Time like [math]((Δs)^2 > 0)[/math] : The time difference is larger than the spatial separation → Causal relationship possible (one event can influence the other).
– Spacelike [math]((Δs)^2 < 0)[/math] : The spatial separation is greater than the time interval → Causal connection impossible (events are too far apart for a signal to travel between them at the speed of light).
– Light like [math]((Δs)^2 = 0)[/math] : The separation is exactly what a light beam would travel → The events are connected by a light signal.
(i) Proper Time Interval [math]\Delta t_0[/math]
The proper time interval [math]\Delta t_0[/math] is the time measured in the rest frame of an object (where the two events occur at the same position). It is always the shortest time interval between two events.
Using the space-time interval formula, when two events occur at the same position in some frame ([math]Δx = 0[/math]), we get:
[math](\Delta s)^2 = (c \Delta t_0)^2[/math]
Since the space-time interval is invariant, for any other observer moving relative to the event:
[math](c \Delta t)^2 – (\Delta x)^2 = (c \Delta t_0)^2[/math]
Rearranging:
[math]\Delta t_0 = \frac{\Delta t}{\gamma}[/math]
Where:
[math]\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}[/math]
This means time intervals measured in a moving frame appear longer than in the rest frame.
(j) Time Dilation
Time dilation means that a moving clock ticks more slowly compared to a stationary clock. If a time interval [math][/math] is measured in the clock’s own rest frame, then for an observer moving relative to the clock, the time interval is:
[math]Δt = γΔt_0[/math]
Where γ is the Lorentz factor:
[math]\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}[/math]
⇒ Consequences of Time Dilation
Moving clocks tick slower compared to stationary ones.
If an astronaut travels close to ccc, they experience less time than people on Earth (this is the basis of the Twin Paradox).
⇒ Experimental Verification:
Muons (particles created in the upper atmosphere) should decay before reaching Earth. However, due to time dilation, they live longer and reach the ground.
Atomic clocks on fast-moving airplanes tick slower than stationary ones on Earth, confirmed by experiments.
⇒ Proper Length [math]L_o[/math] and Length Contraction
The proper length [math]L_o[/math] is the length of an object measured in its own rest frame (where it is stationary). Due to length contraction, a moving observer measures a shorter length:
[math]L = L_o \sqrt{1 – \frac{v^2}{c^2}}[/math]
Where:
– [math]L_o[/math] is the rest length (the length measured when the object is at rest).
– L is the contracted length seen by a moving observer.
⇒ Consequences of Length Contraction
Objects moving close to ccc shrink in the direction of motion.
Only applies along the direction of motion (not perpendicular).
Just like time dilation, it has been experimentally verified using fast-moving particles.
(k) Length Contraction
Length contraction states that an object moving relative to an observer appears shorter along the direction of motion. This effect is a direct consequence of Lorentz transformations in special relativity.
The contracted length L is given by:
[math]L = \frac{L_0}{γ}[/math]
Where:
– [math]L_o[/math] = proper length (measured in the object’s rest frame)
– L = length measured by an observer in motion relative to the object
– γ = Lorentz factor:
[math]\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}[/math]
⇒ Observations
Only the length along the direction of motion contracts; perpendicular dimensions remain unchanged.
As [math]v → c, γ → ∞[/math], so the object appears infinitely contracted
Length contraction is not “real” deformation—it’s a coordinate transformation effect.
Figure 8 Length Contraction
⇒ Example
Suppose a spaceship is 100 m long in its rest frame. If it moves at 8c relative to an observer on Earth, the observer measures:
[math]L = \frac{L_0}{\sqrt{1 – \frac{v^2}{c^2}}} \\
L = \frac{100}{\sqrt{1 – (0.8)^2}} \\
L = \frac{100}{\sqrt{0.36}} \\
L = \frac{100}{0.6} \\
L 60 \, \text{m}[/math]
So, the moving spaceship appears shortened to 60 m.
(l) The Relativity of Simultaneity
In special relativity, simultaneity is relative—two events that are simultaneous in one reference frame may not be simultaneous in another moving frame.
Imagine two lightning strikes at positions A and B.
– In an observer’s rest frame, the lightning strikes appear simultaneous.
– But to an observer moving relative to A and B, one strike happens before the other due to time dilation effects.
This happens because the order of events depends on the frame of reference and is described mathematically by the Lorentz transformation:
[math]t’ = \gamma \left( t – \frac{vx}{c^2} \right)[/math]
Where:
– t′ = event time in the moving frame
– t = event time in the stationary frame
– v = velocity of moving frame
– x = event position in stationary frame
Figure 9 Relativity of simultaneous
Conclusion
No absolute simultaneity—simultaneous events in one frame are not necessarily simultaneous in another.
Time measurements depend on the observer’s motion.
(m) Space-Time Diagrams
A space-time diagram is a visual representation of events in Minkowski space. The axes are:
– Vertical axis: Time (ct)
– Horizontal axis: Space (x)
1. World Line: The path of an object through space-time.
2. Light Cone: The region that represents all possible signals traveling at the speed of light.
3. Simultaneity Lines: These show events that appear simultaneous in a given frame.
Figure 10 Space – time diagram
(n) Angle Between the World Line and Time Axis
If a particle moves at velocity v, the angle θ between its world line and the time axis is:
[math]tan⁡θ = \frac{v}{c}[/math]
– If v = 0, [math]θ = 0^0[/math] → The object is at rest.
– If v = c, [math]θ = 45^0[/math] → The object moves at the speed of light.
– If v > c, [math]θ > 45^0[/math] (not possible for massive objects).
This shows that the faster an object moves, the more tilted its world line becomes.
(o) Muon Decay Experiment: Evidence for Time Dilation & Length Contraction
Muons are unstable subatomic particles created when cosmic rays strike the Earth’s atmosphere.
– In their rest frame, muons have a lifetime of 2 μs before decaying.
– They are produced at an altitude of about 10 km, moving at 99c.
⇒ Problem
Without relativistic effects, a muon should travel:
[math]d = v \times t \\
d = 0.99 \times (2.2 \times 10^{-6}) \\
d = 0.66 \, \text{km}[/math]
This means they should decay before reaching Earth.
Figure 11 Muon decay
⇒ Solution Using Time Dilation
In the Earth’s frame, the muon’s lifetime is dilated:
[math]Δt = γΔt_0[/math]
Where:
[math]\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}[/math]
Thus, the observed lifetime:
[math]\Delta t = 7.09 \times 2.2 \, \mu\text{s} \\
\Delta t = 15.6 \, \mu\text{s} [/math]
New travel distance:
[math]d = (0.99) \times (15.6 \times 10^{-6}) \\
d = 10 \, \text{km}[/math]
Time dilation explains why muons reach Earth:
Solution Using Length Contraction
In the muon’s frame, the Earth appears closer due to length contraction:
[math]L = \frac{L_0}{\gamma} \\
L = \frac{10}{7.09} \\
L \approx 1.41 \, \text{km}[/math]
Since muons only need to travel 1.41 km, they reach Earth before decaying.