How Do Different Observers Describe the Same Event?

Imagine two observers.

Observer S is standing on Earth.

Observer S′ is travelling in a spacecraft at constant velocity:

v

relative to S.

An event occurs somewhere in space.

Observer S describes it using coordinates:

(x, y, z, t)

Observer S′ describes the same event using:

(x′, y′, z′, t′)

The central question is:

How are these two sets of coordinates related?

In classical physics, we use:

Galilean transformations.

In Special Relativity, we must use:

Lorentz transformations.

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Coordinate Transformations

A coordinate transformation is a mathematical rule that allows us to convert measurements from one reference frame into:

another reference frame.

For example, two observers may disagree about an object's:

• position
• velocity
• time
• distance

But their measurements must be connected by consistent:

mathematical relationships.

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Setting Up Two Reference Frames

Consider two inertial reference frames:

S and S′

Suppose S′ moves in the positive x-direction relative to S at constant velocity:

v.

At:

t = t′ = 0

the origins coincide:

x = x′ = 0

The axes are aligned.

Motion occurs along the:

x-axis.

This simple arrangement allows us to derive the transformation equations.

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The Galilean Transformation

In classical mechanics, the relationship between positions is:

x′ = x − vt

The other spatial coordinates remain:

y′ = y

z′ = z

Classical physics also assumes:

t′ = t

This means:

time is absolute.

Every observer agrees on the same time interval.

---

Understanding x′ = x − vt

Suppose S′ moves to the right at:

10 m/s.

After:

5 s

its origin has travelled:

vt = 50 m

relative to S.

If an object is at:

x = 80 m

in S, then S′ assigns:

x′ = 80 − 50

x′ = 30 m

This works extremely well at:

ordinary speeds.

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The Classical Assumption About Time

The most important Galilean assumption is:

t′ = t

If 10 seconds pass for S, then:

10 seconds

also pass for S′.

Time is treated as completely independent of:

motion.

This idea was central to classical Newtonian physics.

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The Problem with Light

Now suppose S observes a pulse of light travelling in the positive x-direction.

Its speed is:

c.

Classical velocity transformation would predict:

u′ = u − v

Therefore:

c′ = c − v

If S′ moved toward the light, classical physics would predict a different measured:

light speed.

But this conflicts with the central postulate of Special Relativity.

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Einstein's Second Postulate

Albert Einstein proposed that:

the speed of light in vacuum is the same for all inertial observers.

Therefore:

c′ = c

not:

c′ = c − v

This means the Galilean transformation cannot be the correct transformation at:

relativistic speeds.

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What Must Change?

The Galilean transformation assumes:

space changes between frames

but:

time does not.

Special Relativity requires something fundamentally different.

To keep the speed of light invariant:

both space and time coordinates must transform.

This leads to the:

Lorentz transformations.

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Assumptions Behind the Lorentz Transformations

The derivation begins with several important assumptions.

1. Principle of Relativity

The laws of physics are the same in all:

inertial reference frames.

There is no preferred inertial frame.

2. Constancy of the Speed of Light

All inertial observers measure the same vacuum light speed:

c.

3. Homogeneity of Space and Time

The laws of physics do not depend on:

where or when an experiment occurs.

4. Isotropy of Space

Physics does not fundamentally depend on:

direction.

5. Linearity

For inertial frames moving uniformly relative to one another, the coordinate transformation is taken to be:

linear.

This preserves uniform motion.

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Why Assume a Linear Transformation?

Suppose an object moves at constant velocity in one inertial frame.

It should also move uniformly in another:

inertial frame.

If the transformation were strongly nonlinear, uniform motion in one frame could become:

accelerated motion

in another.

That would conflict with the equivalence of inertial frames.

Therefore, we seek transformations that are:

linear in x and t.

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Start with a General Form

Because S′ moves along the x-axis, suppose:

x′ = A(x − vt)

where A is a factor that may depend on:

v.

Why use:

x − vt?

Because the origin of S′ satisfies:

x = vt.

At the S′ origin:

x′ = 0

so:

x − vt = 0.

This ensures the moving origin is correctly described.

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The Transformation for Time

Unlike Galilean relativity, we cannot simply assume:

t′ = t.

Instead, time must also depend on position.

We eventually obtain:

t′ = A(t − vx/c²)

The same factor A appears in both transformations.

We now need to determine:

what A must be.

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Use a Pulse of Light

Imagine that when the two origins coincide:

t = t′ = 0

a pulse of light is emitted from the common origin.

According to observer S:

x = ct

According to observer S′:

x′ = ct′

because both observers must measure the same:

speed of light c.

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Substitute the Light Path

We proposed:

x′ = A(x − vt)

and:

t′ = A(t − vx/c²)

For light:

x = ct

Substitute into the position equation:

x′ = A(ct − vt)

Factor:

x′ = At(c − v)

Now substitute into the time equation:

t′ = A(t − vct/c²)

Simplify:

t′ = At(1 − v/c)

Multiplying by c:

ct′ = At(c − v)

Therefore:

x′ = ct′

as required.

The transformation preserves:

the speed of light.

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Finding the Factor A

We still need to determine:

A.

The principle of relativity requires symmetry between:

S and S′.

If S′ moves at velocity v relative to S, then S moves at velocity:

−v

relative to S′.

Therefore, the inverse transformation must have the same form:

x = A(x′ + vt′)

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Substitute the Forward Transformations

Start with:

x = A(x′ + vt′)

Substitute:

x′ = A(x − vt)

and:

t′ = A(t − vx/c²)

Then:

x = A[A(x − vt) + vA(t − vx/c²)]

Factor out A²:

x = A²[x − vt + vt − v²x/c²]

The middle terms cancel:

−vt + vt = 0

Therefore:

x = A²x(1 − v²/c²)

Divide by x:

1 = A²(1 − v²/c²)

So:

A² = 1/(1 − v²/c²)

Therefore:

A = 1/√(1 − v²/c²)

This factor is called:

the Lorentz factor.

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The Lorentz Factor

The Lorentz factor is represented by the Greek letter:

γ

and is defined as:

γ = 1/√(1 − v²/c²)

Therefore:

A = γ

and our transformations become:

x′ = γ(x − vt)

and:

t′ = γ(t − vx/c²)

These are the central:

Lorentz transformation equations.

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Complete Lorentz Transformations

For relative motion along the x-axis:

x′ = γ(x − vt)

y′ = y

z′ = z

t′ = γ(t − vx/c²)

where:

γ = 1/√(1 − v²/c²)

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What Do These Equations Mean?

The equations show that measurements of:

space and time are interconnected.

Notice:

x′ depends on x and t

and:

t′ depends on t and x.

This means different inertial observers do not simply disagree about:

position.

They can also disagree about:

time.

This is one of the deepest differences between classical and relativistic physics.

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The Inverse Lorentz Transformations

To transform from S′ back to S, replace:

v with −v.

Therefore:

x = γ(x′ + vt′)

and:

t = γ(t′ + vx′/c²)

The transverse coordinates remain:

y = y′

z = z′

The symmetry reflects the principle that neither inertial frame is:

preferred.

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Why γ Matters

The Lorentz factor controls the size of:

relativistic effects.

Recall:

γ = 1/√(1 − v²/c²)

When:

v ≪ c

then:

v²/c² ≈ 0

so:

γ ≈ 1.

Therefore, Lorentz transformations become very close to:

Galilean transformations.

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Values of the Lorentz Factor

At low speeds:

γ ≈ 1

Near the speed of light:

γ increases rapidly.

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Recovering the Galilean Transformation

Consider:

x′ = γ(x − vt)

At low speeds:

γ ≈ 1

Therefore:

x′ ≈ x − vt

which is exactly the Galilean position transformation.

Now consider:

t′ = γ(t − vx/c²)

At low speeds:

γ ≈ 1

and:

vx/c²

is extremely small.

Therefore:

t′ ≈ t

which is the classical assumption of:

absolute time.

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Classical Physics Is an Approximation

This is an important scientific idea.

Special Relativity does not mean that classical mechanics is:

useless or completely wrong.

Instead:

Galilean transformations are the low-speed approximation of Lorentz transformations.

When:

v ≪ c

the difference becomes negligible.

This is why classical mechanics works extremely well for:

• cars
• bicycles
• aircraft
• falling objects
• most engineering systems

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Galilean vs Lorentz Transformations

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Worked Example 1: Calculate γ

A spacecraft travels at:

0.60c

Calculate the Lorentz factor.

γ = 1/√(1 − v²/c²)

Since:

v = 0.60c

then:

v²/c² = 0.36

Therefore:

γ = 1/√(1 − 0.36)

γ = 1/√0.64

γ = 1.25

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Worked Example 2: Transforming Position

Suppose:

v = 0.60c

and an event occurs at:

x = 9.0 × 10⁸ m

at:

t = 4.0 s

We know:

γ = 1.25

Use:

x′ = γ(x − vt)

Calculate:

vt = (0.60)(3.0 × 10⁸)(4.0)

vt = 7.2 × 10⁸ m

Therefore:

x′ = 1.25(9.0 × 10⁸ − 7.2 × 10⁸)

x′ = 1.25(1.8 × 10⁸)

x′ = 2.25 × 10⁸ m

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Worked Example 3: Transforming Time

Use the same event:

x = 9.0 × 10⁸ m

t = 4.0 s

v = 0.60c

γ = 1.25

Use:

t′ = γ(t − vx/c²)

First calculate:

vx/c²

= (0.60c)(9.0 × 10⁸)/c²

Using:

c = 3.0 × 10⁸ m/s

this becomes:

1.8 s

Therefore:

t′ = 1.25(4.0 − 1.8)

t′ = 1.25(2.2)

t′ = 2.75 s

So S′ assigns the event:

x′ = 2.25 × 10⁸ m

t′ = 2.75 s

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Same Event, Different Coordinates

Observer S describes the event as:

(9.0 × 10⁸ m, 4.0 s)

Observer S′ describes it as:

(2.25 × 10⁸ m, 2.75 s)

These are not different:

events.

They are different coordinate descriptions of:

the same event.

This is similar to two maps assigning different coordinates to the same location.

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Worked Example 4: Transforming a Light Pulse

Suppose a light pulse travels according to:

x = ct

Take:

t = 2.0 s

Then:

x = 6.0 × 10⁸ m

Suppose S′ moves at:

0.60c.

Then:

γ = 1.25

Position:

x′ = γ(x − vt)

x′ = 1.25[(6.0 × 10⁸) − (0.60)(3.0 × 10⁸)(2.0)]

x′ = 3.0 × 10⁸ m

Time:

t′ = γ(t − vx/c²)

t′ = 1.0 s

Therefore:

x′/t′ = 3.0 × 10⁸ m/s

So S′ also measures:

c.

This is exactly what the Lorentz transformations must accomplish.

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Galilean Transformation of the Same Light Pulse

Classically:

x′ = x − vt

Using the same example:

x′ = 6.0 × 10⁸ − 3.6 × 10⁸

x′ = 2.4 × 10⁸ m

Galilean physics also says:

t′ = 2.0 s

Therefore:

u′ = x′/t′

u′ = 1.2 × 10⁸ m/s

That is:

0.40c

But Special Relativity requires:

c.

This demonstrates why Galilean transformations fail for:

light and relativistic motion.

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Space and Time Become Mixed

The Lorentz equations are:

x′ = γ(x − vt)

t′ = γ(t − vx/c²)

Notice that:

position transformation contains time

and:

time transformation contains position.

Space and time can no longer be treated as completely:

independent quantities.

They form a unified structure called:

spacetime.

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Lorentz Transformations and Time Dilation

Time dilation follows from the:

Lorentz transformations.

Consider a clock at rest in S′.

For that clock:

Δx′ = 0.

Applying the inverse transformation leads to:

Δt = γΔt′

If the clock measures proper time:

Δτ = Δt′

then:

Δt = γΔτ

This is the familiar:

time dilation equation.

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Lorentz Transformations and Length Contraction

Length contraction also follows from the transformations.

Suppose an object has proper length:

L₀

in its own rest frame.

An observer who sees the object moving must measure the positions of both ends:

simultaneously in that observer's frame.

Applying the Lorentz transformation gives:

L = L₀/γ

This is the:

length contraction equation.

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Lorentz Transformations and Simultaneity

Suppose two events are simultaneous in S:

Δt = 0

but occur at different positions:

Δx ≠ 0.

Then:

Δt′ = γ(Δt − vΔx/c²)

so:

Δt′ = −γvΔx/c²

Therefore:

Δt′ ≠ 0

in general.

This produces the:

relativity of simultaneity.

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One Transformation, Several Effects

The Lorentz transformations explain:

time dilation

length contraction

relativity of simultaneity

relativistic velocity addition

These are not independent assumptions.

They are consequences of the same underlying:

transformation between inertial frames.

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Relativistic Velocity Transformation

Galilean physics predicts:

u′ = u − v

But Special Relativity gives:

u′ = (u − v)/(1 − uv/c²)

This equation ensures that if:

u = c

then:

u′ = c.

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Example: Light Speed

Suppose:

u = c

and:

v = 0.80c.

Then:

u′ = (c − 0.80c)/(1 − (c)(0.80c)/c²)

u′ = 0.20c/(1 − 0.80)

u′ = 0.20c/0.20

u′ = c

The moving observer still measures:

the speed of light as c.

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Example: Two Spacecraft

Suppose spacecraft A moves at:

0.80c

relative to Earth.

Spacecraft B moves in the same direction at:

0.60c

relative to Earth.

Classically, A would measure B moving at:

0.20c

relative to it.

Relativistically:

u′ = (0.60c − 0.80c)/(1 − 0.60 × 0.80)

u′ = −0.20c/0.52

u′ ≈ −0.385c

The magnitude of the relative velocity is:

0.385c.

At relativistic speeds, velocities do not simply:

subtract classically.

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Why Can't Massive Objects Reach c?

Consider:

γ = 1/√(1 − v²/c²)

As:

v → c

then:

1 − v²/c² → 0

Therefore:

γ → ∞

Many relativistic quantities involving γ grow without bound as a massive object's speed approaches:

c.

This is one mathematical indication that an object with nonzero rest mass cannot be accelerated to:

the speed of light.

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The Spacetime Interval

Lorentz transformations change measurements of:

space and time.

But they preserve an important quantity called the:

spacetime interval.

For two events:

Δs² = c²Δt² − Δx² − Δy² − Δz²

All inertial observers calculate the same:

Δs².

This is called:

Lorentz invariance.

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Compare with Distance in Ordinary Geometry

Imagine rotating coordinate axes on a sheet of paper.

The x and y coordinates of a point change.

But the distance:

r² = x² + y²

does not.

Similarly, Lorentz transformations change:

space and time coordinates

while preserving the:

spacetime interval.

This provides a useful geometric way to understand:

Special Relativity.

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Minkowski Spacetime

Hermann Minkowski developed a geometric interpretation of Special Relativity.

Instead of treating space and time separately, events are represented in:

four-dimensional spacetime.

Coordinates can be written:

(ct, x, y, z).

Lorentz transformations describe how different inertial observers assign coordinates within this:

spacetime.

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Galilean Spacetime vs Relativistic Spacetime

Galilean View

Space:

relative

Time:

absolute

Transformation:

x′ = x − vt

t′ = t

Relativistic View

Space:

frame-dependent

Time:

frame-dependent

Transformation:

x′ = γ(x − vt)

t′ = γ(t − vx/c²)

Invariant quantity:

spacetime interval

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Why the Lorentz Factor Appears Everywhere

You have already encountered γ in:

time dilation

Δt = γΔτ

and:

length contraction

L = L₀/γ

Now we see where it comes from.

It is not an arbitrary correction added to equations.

It emerges from requiring the coordinate transformations to satisfy:

• the principle of relativity
• invariance of c
• symmetry between inertial frames

The Lorentz factor is therefore built into the:

geometry of spacetime.

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At Everyday Speeds

Suppose a car travels at:

30 m/s.

Then:

v/c ≈ 10⁻⁷

so:

γ ≈ 1

and:

vx/c²

is extremely small for ordinary distances.

Therefore:

x′ ≈ x − vt

and:

t′ ≈ t.

Galilean transformations are therefore perfectly adequate for:

most everyday situations.

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At Relativistic Speeds

Suppose:

v = 0.90c

Then:

γ ≈ 2.294

Now γ is far from:

1.

The differences between Lorentz and Galilean transformations become:

very large.

At these speeds, classical transformations cannot accurately describe:

space, time or velocity.

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Worked Comparison

Suppose an event occurs at:

x = 3.0 × 10⁸ m

t = 2.0 s

and S′ moves at:

0.80c.

Galilean Transformation

x′ = x − vt

x′ = 3.0 × 10⁸ − (0.80)(3.0 × 10⁸)(2)

x′ = −1.8 × 10⁸ m

and:

t′ = 2.0 s

Lorentz Transformation

For:

v = 0.80c

γ = 1.667

Therefore:

x′ = 1.667(−1.8 × 10⁸)

x′ ≈ −3.0 × 10⁸ m

For time:

t′ = γ(t − vx/c²)

t′ = 1.667(2.0 − 0.80)

t′ ≈ 2.0 s

In this particular event, the transformed time happens to remain 2.0 s, while the transformed position differs substantially. Other event coordinates generally produce differences in both.

The important point is that the two theories make:

different quantitative predictions.

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Choosing the Correct Transformation

Use Galilean transformations when:

• speeds are much smaller than c
• relativistic precision is unnecessary
• classical mechanics provides a sufficient approximation

Use Lorentz transformations when:

• speeds are a significant fraction of c
• light propagation is important
• relativistic precision is required
• studying particle physics or high-energy processes

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Common Misconception: Lorentz Transformations Were Invented Just to Fix Time Dilation

No.

Lorentz transformations provide the fundamental coordinate relationship between:

inertial frames in Special Relativity.

Time dilation is one:

consequence.

So are:

• length contraction
• relativity of simultaneity
• relativistic velocity addition

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Common Misconception: Galilean Transformations Are Wrong

They are not useless or meaningless.

They are an excellent:

low-speed approximation.

When:

v/c → 0

Lorentz transformations approach:

Galilean transformations.

This is called the:

correspondence principle.

A more general theory should reproduce the successful predictions of an older theory in the conditions where the older theory works.

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Common Misconception: γ Changes the Speed of Light

No.

The Lorentz factor helps ensure that different inertial observers all measure:

the same c.

It changes how their space and time coordinates are:

related.

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Common Misconception: Time Is Universal

Galilean transformations assume:

t′ = t.

Lorentz transformations show:

t′ = γ(t − vx/c²).

Therefore, the time assigned to a distant event depends on:

• the event's position
• relative velocity
• reference frame

Time is not universally:

absolute.

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Common Misconception: Different Coordinates Mean Different Events

No.

Two observers can assign different:

x and t coordinates

to the same event.

The event itself is the same physical occurrence.

Coordinates depend on:

reference frame.

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Check Your Understanding

1. What is a coordinate transformation?

2. State the Galilean transformation for position.

3. What assumption does Galilean relativity make about time?

4. Why does the Galilean transformation fail for light?

5. State Einstein's two postulates.

6. List three assumptions used when deriving Lorentz transformations.

7. Why is the transformation assumed to be linear?

8. State the Lorentz transformation for position.

9. State the Lorentz transformation for time.

10. Define the Lorentz factor.

11. Calculate γ for v = 0.60c.

12. Explain why γ approaches 1 at low speeds.

13. Explain how the Lorentz transformations reduce to Galilean transformations when v ≪ c.

14. Why does time transformation contain a position term?

15. Show that a light pulse travelling at c in S also travels at c in S′.

16. Explain how time dilation follows from Lorentz transformations.

17. Explain how length contraction follows from Lorentz transformations.

18. Explain how relativity of simultaneity follows from Lorentz transformations.

19. What quantity remains invariant under Lorentz transformations?

20. Why do physicists use Galilean transformations for everyday motion but Lorentz transformations for relativistic motion?

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Key Terms

• Coordinate transformation: Mathematical relationship connecting coordinates assigned by different reference frames.
• Galilean transformation: Classical transformation assuming absolute time.
• Lorentz transformation: Relativistic transformation connecting space and time coordinates between inertial frames.
• Inertial reference frame: Non-accelerating frame in which a free object moves at constant velocity.
• Lorentz factor (γ): Factor 1/√(1 − v²/c²) controlling the magnitude of relativistic effects.
• Principle of relativity: Laws of physics have the same form in all inertial frames.
• Constancy of light speed: All inertial observers measure the same vacuum light speed c.
• Linearity: Property in which transformed coordinates depend linearly on the original coordinates.
• Homogeneity: Principle that the laws of physics do not depend on absolute position or time.
• Isotropy: Principle that physical laws do not depend on spatial direction.
• Spacetime: Unified four-dimensional description of space and time.
• Spacetime interval: Lorentz-invariant separation between events.
• Lorentz invariance: Property that fundamental physical laws and the spacetime interval have the appropriate invariant form under Lorentz transformations.
• Inverse transformation: Transformation converting coordinates back to the original frame.
• Correspondence principle: Requirement that a newer theory reproduce the successful predictions of an older theory in the older theory's valid limit.

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Key Takeaways

• Coordinate transformations connect measurements made in different reference frames.
• Classical mechanics uses Galilean transformations.
• Galilean transformations assume t′ = t, meaning time is absolute.
• Classical velocity addition would predict different measured speeds of light for different observers.
• This conflicts with Einstein's postulate that all inertial observers measure the same vacuum speed of light, c.
• Special Relativity therefore requires Lorentz transformations.
• The derivation assumes the principle of relativity, invariance of c, homogeneity, isotropy and linearity.
• For relative motion along the x-axis, x′ = γ(x − vt).
• The time transformation is t′ = γ(t − vx/c²).
• The Lorentz factor is γ = 1/√(1 − v²/c²).
• The Lorentz factor arises from requiring symmetry between inertial frames while preserving c.
• Lorentz transformations mix space and time coordinates.
• This mixing is a fundamental feature of spacetime.
• Time dilation follows from Lorentz transformations.
• Length contraction follows from Lorentz transformations.
• Relativity of simultaneity follows from Lorentz transformations.
• Relativistic velocity addition also follows from the same framework.
• Lorentz transformations preserve the spacetime interval.
• At low speeds, γ ≈ 1 and vx/c² ≈ 0.
• Therefore, Lorentz transformations reduce approximately to Galilean transformations.
• Galilean physics remains an excellent approximation for ordinary speeds.
• At speeds approaching c, the differences become significant and Lorentz transformations are essential.
• The Lorentz factor is not an arbitrary correction—it is a fundamental consequence of the geometry and symmetry of Special Relativity.