Measuring Separation in Spacetime

Imagine two events:

Event A: a spacecraft leaves Earth.

Event B: the spacecraft sends a signal several seconds later.

How far apart are these events?

In ordinary geometry, we might calculate only the difference in:

position.

But in relativity, the events are separated in both:

space and time.

To describe their separation in spacetime, physicists use the:

space-time interval, more commonly written spacetime interval.

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What Is the Spacetime Interval?

The spacetime interval is a quantity that combines the spatial separation and time separation between two events.

For motion in one spatial dimension:

Δs² = c²Δt² − Δx²

where:

• Δs² = spacetime interval squared
• c = speed of light
• Δt = time separation between the events
• Δx = spatial separation between the events

The speed of light is approximately:

c = 3.00 × 10⁸ m/s

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Three Spatial Dimensions

In three-dimensional space, the equation becomes:

Δs² = c²Δt² − Δx² − Δy² − Δz²

This combines:

three spatial dimensions

with:

one time dimension.

It is one of the fundamental mathematical relationships of:

Special Relativity.

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A Note About Sign Conventions

Some textbooks write the spacetime interval as:

Δs² = Δx² + Δy² + Δz² − c²Δt²

instead.

This is not a different physical theory.

It is simply a different:

sign convention.

In these notes, we will use:

Δs² = c²Δt² − Δx²

for one-dimensional motion.

The important thing is to remain:

consistent.

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Finding Δx and Δt

Suppose two events have coordinates:

Event A: (x₁, t₁)

Event B: (x₂, t₂)

Then:

Δx = x₂ − x₁

and:

Δt = t₂ − t₁.

These differences are then substituted into:

Δs² = c²Δt² − Δx².

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Why Multiply Time by c?

Space is normally measured in:

metres.

Time is normally measured in:

seconds.

We cannot directly subtract seconds squared from metres squared.

But:

cΔt

has units of distance.

Since:

c = m/s

then:

cΔt = (m/s)(s) = m.

Therefore:

c²Δt²

has units of:

m².

Now both terms in the spacetime interval have compatible units.

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Ordinary Distance

In ordinary three-dimensional Euclidean space, the distance between two points is:

d² = Δx² + Δy² + Δz².

For a simple two-dimensional example:

d² = Δx² + Δy².

This is based on the:

Pythagorean theorem.

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Spacetime Interval vs Ordinary Distance

The spacetime interval is different.

Ordinary distance considers separation in:

space.

The spacetime interval considers separation in:

space and time.

Ordinary distance:

d² = Δx² + Δy² + Δz²

Spacetime interval:

Δs² = c²Δt² − Δx² − Δy² − Δz²

Notice the important:

minus signs.

These are a fundamental feature of:

Minkowski spacetime.

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Why Isn't It Just the Pythagorean Theorem?

In ordinary geometry:

distance² = x² + y².

But spacetime does not have ordinary Euclidean geometry.

Its geometry is:

Minkowskian.

Time contributes differently from:

space.

This mathematical difference is responsible for many important features of Special Relativity.

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A Convenient Unit: Light-Seconds

Calculations become easier if distance is measured in:

light-seconds.

One light-second is the distance travelled by light in one second:

1 light-second ≈ 3.00 × 10⁸ m.

If time is measured in seconds and distance in light-seconds, we can effectively use:

c = 1 light-second per second.

Then:

cΔt = Δt

when expressed in corresponding light-second units.

This makes many introductory calculations much simpler.

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Worked Example 1: Events at the Same Position

Two events occur at the same position.

The first occurs at:

t₁ = 2 s

and the second at:

t₂ = 7 s.

Therefore:

Δt = 7 − 2 = 5 s

and:

Δx = 0.

Using:

Δs² = c²Δt² − Δx²

we get:

Δs² = c²(5)² − 0

Δs² = 25c².

If distance is expressed in light-seconds:

Δs² = 25 light-seconds².

Therefore:

Δs = 5 light-seconds.

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Worked Example 2: Separation in Space and Time

Event A occurs at:

x = 0 light-seconds, t = 0 s.

Event B occurs at:

x = 3 light-seconds, t = 5 s.

Therefore:

Δx = 3 light-seconds

and:

cΔt = 5 light-seconds.

Use:

Δs² = c²Δt² − Δx².

So:

Δs² = 5² − 3²

Δs² = 25 − 9

Δs² = 16 light-seconds².

Therefore:

Δs = 4 light-seconds.

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What Does This Answer Mean?

Notice something interesting.

The time separation corresponds to:

5 light-seconds.

The spatial separation is:

3 light-seconds.

But the spacetime interval is:

4 light-seconds.

The interval is not simply the spatial distance or the time difference.

It represents a particular combination of:

space and time separation.

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Worked Example 3: Light

Suppose a flash of light travels:

4 light-seconds

during:

4 seconds.

Therefore:

Δx = 4 light-seconds

and:

cΔt = 4 light-seconds.

Then:

Δs² = 4² − 4²

Δs² = 16 − 16

Δs² = 0.

The spacetime interval is:

zero.

This is called a:

lightlike or null interval.

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Can Two Different Events Have an Interval of Zero?

Yes.

This is one of the surprising features of spacetime geometry.

Two different events can have:

Δs² = 0

if they can be connected by:

light.

The events are not at the same place or time.

Instead:

c²Δt² = Δx².

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Three Types of Spacetime Interval

Using our sign convention:

Δs² = c²Δt² − Δx²

there are three possibilities.

Timelike

Δs² > 0

Lightlike

Δs² = 0

Spacelike

Δs² < 0

Each has an important physical meaning.

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Timelike Intervals

A timelike interval occurs when:

c²Δt² > Δx².

There is enough time for something travelling slower than light to move between the events.

Therefore, the events can potentially be connected by:

a massive object or causal signal.

On a Minkowski diagram, the second event lies:

inside the light cone.

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Example of a Timelike Interval

Suppose:

cΔt = 10 light-seconds

and:

Δx = 6 light-seconds.

Then:

Δs² = 10² − 6²

Δs² = 100 − 36

Δs² = 64 light-seconds².

Since:

Δs² > 0

the separation is:

timelike.

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Lightlike Intervals

A lightlike interval occurs when:

c²Δt² = Δx².

Therefore:

Δs² = 0.

Only something travelling at exactly:

c

can connect the two events.

Light follows these paths through spacetime.

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Example of a Lightlike Interval

Suppose:

cΔt = 7 light-seconds

and:

Δx = 7 light-seconds.

Then:

Δs² = 7² − 7²

Δs² = 49 − 49

Δs² = 0.

The events are:

lightlike separated.

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Spacelike Intervals

A spacelike interval occurs when:

Δx² > c²Δt².

Using our sign convention:

Δs² < 0.

There is not enough time for light to travel between the events.

Therefore, the events cannot be causally connected by any signal travelling at or below:

c.

On a Minkowski diagram, the second event lies:

outside the light cone.

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Example of a Spacelike Interval

Suppose:

cΔt = 4 light-seconds

and:

Δx = 7 light-seconds.

Then:

Δs² = 4² − 7²

Δs² = 16 − 49

Δs² = −33 light-seconds².

Since:

Δs² < 0,

the separation is:

spacelike.

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What Does a Negative Interval Mean?

A negative value of Δs² does not mean that the physical separation is somehow a "negative distance."

It tells us that the spatial separation dominates the temporal separation.

In our chosen sign convention:

negative Δs² → spacelike separation.

Often it is better to leave the result as:

Δs² = −33 light-seconds²

rather than trying to take an ordinary square root of a negative number.

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The Light Cone and Intervals

The three types of interval correspond directly to regions on a Minkowski diagram.

Inside the light cone:

timelike

On the light cone:

lightlike

Outside the light cone:

spacelike

This makes Minkowski diagrams particularly useful for interpreting:

spacetime intervals.

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What Does Invariant Mean?

The most important property of the spacetime interval is that it is:

invariant.

An invariant quantity has the same value for all:

inertial observers.

Two observers may disagree about:

Δx

and:

Δt.

But when each calculates:

c²Δt² − Δx²,

they obtain the:

same result.

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Observer S and Observer S′

Suppose Observer S measures:

Δx

and:

Δt.

Observer S′ moves relative to S and measures:

Δx′

and:

Δt′.

Special Relativity tells us:

c²Δt² − Δx² = c²Δt′² − Δx′².

Therefore:

Δs² = Δs′².

This is the invariance of the:

spacetime interval.

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Why Do the Measurements Differ?

The observers are moving relative to one another.

Lorentz transformations tell us:

Δx′ = γ(Δx − vΔt)

and:

Δt′ = γ(Δt − vΔx/c²).

Therefore, changing reference frames changes:

space measurements

and:

time measurements.

But these changes occur in exactly the right way to preserve:

the spacetime interval.

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A Geometrical Analogy

Imagine measuring a line on a sheet of paper.

One person uses horizontal and vertical coordinates:

x and y.

Another uses rotated coordinates:

x′ and y′.

They may disagree about the individual coordinate differences.

But they agree on:

d² = Δx² + Δy².

The distance remains unchanged.

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The Spacetime Version

Something similar happens in Special Relativity.

Different observers use:

x, t

and:

x′, t′.

Their measurements of space and time differ.

But:

c²Δt² − Δx²

remains unchanged.

Lorentz transformations therefore preserve:

spacetime intervals.

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Ordinary Rotations vs Lorentz Transformations

There is an important difference.

Ordinary rotations preserve:

Δx² + Δy².

Lorentz transformations preserve:

c²Δt² − Δx².

This difference reflects the unusual geometry of:

spacetime.

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Worked Example: Two Observers

Suppose Observer S measures:

Δt = 5 s

and:

Δx = 3 light-seconds.

Then:

Δs² = 5² − 3²

Δs² = 16 light-seconds².

Now suppose Observer S′ measures:

Δt′ = 4 s

and:

Δx′ = 0.

Then:

Δs′² = 4² − 0²

Δs′² = 16 light-seconds².

Both observers obtain:

the same spacetime interval.

Their measurements of distance and time differ, but the interval is:

invariant.

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Why Was Δx′ = 0?

In the previous example, Observer S′ is in a frame where the two events occur at:

the same location.

Therefore:

Δx′ = 0.

The time between those events in that frame is:

proper time.

This gives us an important connection between:

spacetime intervals and time dilation.

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Spacetime Interval and Proper Time

For timelike-separated events, there is a reference frame in which:

Δx = 0.

In that frame:

Δs² = c²Δτ²,

where:

Δτ

is the proper time.

Therefore:

Δs = cΔτ.

This connects the geometry of spacetime directly to:

time measured by a clock moving between the events.

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Example: Proper Time

Suppose:

Δs² = 36 light-seconds².

Then:

Δs = 6 light-seconds.

In the frame where the events occur at the same position:

cΔτ = 6 light-seconds.

Therefore:

Δτ = 6 s.

The proper time between the events is:

6 seconds.

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Spacelike Intervals and Proper Distance

For spacelike-separated events, there is a reference frame in which the two events occur:

simultaneously.

Therefore:

Δt = 0.

The spatial separation measured in that frame is closely related to the invariant interval and is often called the:

proper distance.

So spacetime intervals connect not only to proper time, but also to:

spatial separation.

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Physical Meaning of Timelike Separation

If two events are timelike separated:

one event can potentially influence the other.

For example:

A spacecraft leaves Earth.

Later, the astronaut sends a radio message.

The departure and message transmission can be connected by the astronaut's:

worldline.

The events therefore have a:

timelike relationship.

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Physical Meaning of Lightlike Separation

If two events are lightlike separated:

a light signal can connect them.

For example:

A laser pulse leaves Earth.

Later, it reaches a spacecraft.

The emission and reception events are connected by the worldline of:

light.

Their interval is:

zero.

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Physical Meaning of Spacelike Separation

If two events are spacelike separated:

neither can causally influence the other.

No signal travelling at or below the speed of light has enough time to travel between them.

This is fundamental to the relativistic idea of:

causality.

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Intervals and Causality

The spacetime interval tells us more than "how far apart" two events are.

It tells us something about whether they can be:

causally connected.

This table assumes our chosen sign convention.

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The Order of Timelike Events

Suppose A occurs before B and the events are:

timelike separated.

All inertial observers agree that A occurs:

before B.

They may disagree about the amount of time between them, but not their:

causal order.

This is essential for preserving:

cause and effect.

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The Order of Spacelike Events

Spacelike events behave differently.

Observer S may say:

A happens before B.

Another observer may say:

B happens before A.

A third frame may find that they happen:

simultaneously.

This does not violate causality because spacelike events cannot:

causally influence one another.

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The Order of Lightlike Events

Lightlike-separated events have a fixed causal order.

If a light signal is emitted at A and received at B, all inertial observers agree that:

A precedes B.

They also agree that the connecting signal travels at:

c.

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Why Invariance Matters

Imagine if different observers disagreed about the fundamental spacetime relationship between two events.

One might conclude that the events could be causally connected.

Another might conclude that they could not.

Special Relativity avoids this problem.

Observers may disagree about:

distance and elapsed time separately,

but they agree about the:

spacetime interval and causal classification.

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Spacetime Interval and Minkowski Diagrams

On a Minkowski diagram:

• timelike events lie inside the light cone
• lightlike events lie on the light cone
• spacelike events lie outside the light cone

The spacetime interval gives the mathematical version of what the diagram shows:

geometrically.

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Worked Example: Classify the Interval

Two events have:

Δt = 8 s

and:

Δx = 6 light-seconds.

Using c = 1 light-second per second:

Δs² = 8² − 6²

Δs² = 64 − 36

Δs² = 28 light-seconds².

Since:

Δs² > 0,

the interval is:

timelike.

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Worked Example: Another Classification

Suppose:

Δt = 3 s

and:

Δx = 5 light-seconds.

Then:

Δs² = 3² − 5²

Δs² = 9 − 25

Δs² = −16 light-seconds².

Since:

Δs² < 0,

the interval is:

spacelike.

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Worked Example Using SI Units

Suppose:

Δt = 2.0 s

and:

Δx = 3.0 × 10⁸ m.

Use:

Δs² = c²Δt² − Δx².

Substitute:

Δs² = (3.00 × 10⁸)²(2.0)² − (3.0 × 10⁸)²

First term:

c²Δt² = 3.6 × 10¹⁷ m²

Second term:

Δx² = 9.0 × 10¹⁶ m²

Therefore:

Δs² = 2.7 × 10¹⁷ m².

Since the answer is positive:

the events are timelike separated.

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A Faster Classification Method

You do not always need to calculate the entire interval.

Compare:

Δx

with:

cΔt.

If:

Δx < cΔt

→ timelike

If:

Δx = cΔt

→ lightlike

If:

Δx > cΔt

→ spacelike

This is often the quickest way to determine the:

type of interval.

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Common Misconception: The Interval Is Just Distance

It is not.

Ordinary distance measures separation through:

space.

The spacetime interval combines separation through:

space and time.

That is why two observers can disagree about distance but still agree about:

the spacetime interval.

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Common Misconception: Negative Means Impossible

A negative value of Δs² does not mean the events are impossible.

It means the events are:

spacelike separated

under our sign convention.

Both events can exist perfectly normally.

They simply cannot be causally connected by a signal travelling at or below:

the speed of light.

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Common Misconception: Zero Interval Means Same Event

Not necessarily.

The same event obviously has:

Δx = 0 and Δt = 0.

But two different events connected by light also have:

Δs² = 0.

These events are:

lightlike separated.

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Common Misconception: Every Observer Measures the Same Time

Observers generally do not agree on:

Δt.

They also generally do not agree on:

Δx.

What they agree on is the combination:

c²Δt² − Δx².

This is why the spacetime interval is called:

invariant.

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Common Misconception: Invariant Means Nothing Changes

"Invariant" does not mean every measurement remains unchanged.

Instead:

Δx changes

and:

Δt changes,

while:

Δs² remains unchanged.

The individual coordinates depend on the observer.

The spacetime interval does:

not.

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Connecting the Ideas

The spacetime interval connects several ideas from Special Relativity.

Lorentz transformations

change x and t while preserving Δs².

Time dilation

appears when timelike-separated events are measured in different frames.

Length contraction

is related to different spatial measurements between appropriately chosen simultaneous events.

Relativity of simultaneity

explains why observers disagree about which distant events occur at the same time.

Minkowski diagrams

provide a visual representation of these relationships.

The spacetime interval ties these ideas together mathematically.

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Why the Spacetime Interval Matters

The spacetime interval gives us something all inertial observers can:

agree upon.

Observers can disagree about:

• position
• distance
• elapsed time
• simultaneity

But they agree about:

• the spacetime interval
• whether separation is timelike, lightlike, or spacelike
• the associated causal structure

This makes the spacetime interval one of the central invariant quantities of:

Special Relativity.

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Check Your Understanding

1. Define the spacetime interval.

2. Write the equation for the spacetime interval in one spatial dimension.

3. What do Δx and Δt represent?

4. Why is time multiplied by c in the interval equation?

5. How is a spacetime interval different from ordinary spatial distance?

6. What does it mean for a quantity to be invariant?

7. Two events have Δt = 5 s and Δx = 3 light-seconds. Calculate Δs².

8. Classify the interval in Question 7.

9. Two events have Δt = 4 s and Δx = 4 light-seconds. Calculate Δs².

10. What type of interval is this?

11. Two events have Δt = 2 s and Δx = 6 light-seconds. Calculate Δs².

12. What type of interval is this?

13. What physical connection is possible between timelike-separated events?

14. What can connect two lightlike-separated events?

15. Why can spacelike-separated events not causally influence one another?

16. Explain why two different observers can measure different Δx and Δt values but obtain the same Δs².

17. How is proper time related to a timelike spacetime interval?

18. Where are timelike events located on a Minkowski diagram?

19. Where are spacelike events located?

20. Explain why invariance of the spacetime interval is important to Special Relativity.

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Key Terms

• Spacetime interval: Invariant quantity combining spatial and temporal separation between two events.
• Event: Occurrence at a particular location and time.
• Spatial separation: Difference in position between two events.
• Time separation: Difference in time between two events.
• Invariant: Quantity that has the same value for all inertial observers.
• Minkowski spacetime: Flat four-dimensional spacetime used in Special Relativity.
• Timelike interval: Interval for which a slower-than-light causal connection is possible.
• Lightlike interval: Interval for events that can be connected by light.
• Null interval: Another name for a lightlike interval with Δs² = 0.
• Spacelike interval: Interval for which no causal signal at or below c can connect the events.
• Proper time: Time measured by a clock moving between two timelike-separated events.
• Proper distance: Spatial separation measured in a frame where two spacelike-separated events are simultaneous.
• Lorentz transformation: Transformation connecting coordinates measured in different inertial frames.
• Light cone: Boundary separating different causal regions of spacetime.
• Causality: Principle governing possible cause-and-effect relationships between events.

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Key Takeaways

• The spacetime interval measures separation between events in spacetime rather than simply in space.
• In one spatial dimension, using our sign convention: Δs² = c²Δt² − Δx².
• In three dimensions: Δs² = c²Δt² − Δx² − Δy² − Δz².
• Some textbooks use the opposite sign convention; both conventions describe the same physics when used consistently.
• Ordinary distance and spacetime interval are not the same quantity.
• Multiplying time by c converts a time interval into a distance-like quantity.
• Light-seconds make introductory interval calculations particularly convenient.
• If Δs² > 0, the interval is timelike under our convention.
• If Δs² = 0, the interval is lightlike or null.
• If Δs² < 0, the interval is spacelike.
• Timelike-separated events can potentially be connected by objects or signals travelling below c.
• Lightlike-separated events can be connected by light.
• Spacelike-separated events cannot be causally connected by signals travelling at or below c.
• Different inertial observers generally measure different values of Δx and Δt.
• All inertial observers nevertheless calculate the same spacetime interval.
• Lorentz transformations preserve the spacetime interval.
• For timelike events, the interval is directly related to proper time.
• The interval determines important information about causality.
• On a Minkowski diagram, timelike events lie inside the light cone, lightlike events lie on it, and spacelike events lie outside it.
• The invariance of the spacetime interval provides a common physical quantity on which different inertial observers can agree.
• This makes the spacetime interval one of the fundamental geometrical ideas underlying Special Relativity.