Building Spacetime Diagrams

PHYS 206 – Spring 2014

t

x

The “proper length” (radius) of the circle in any rotated frame of reference is:

Δr = 4

Δr =

4

Δr = 4

But their coordinates do not agree!

y

x

Think of the red line as the “rest” frame (inertial clock with v=0).

Δr = 4

The vector pointing purely in the y-direction is like the worldline of a motionless observer, as measured in their own rest frame!

y

x

A frame of reference “moving” (as measured in the original frame) corresponds to a rotation of the vector.

Δr =

4

Δr = 4

These coordinates are measured by the red observer in the red (motionless) reference frame!

y

x

But if we transform into the blue (moving) reference frame, the vectors look like this:

Δr = 4Δr = 4

These coordinates are measured by the blue observer in the blue (now motionless) reference frame! They are the same as the red coordinates from before (as measured in the red rest frame).

Now, as measured in the blue rest frame, the red frame looks like it is moving (at the same velocity in the opposite direction).

t

x

Δτ = 4

Δτ =

4

Vectors of the same length in hyperbolic spacetime are not the same length “on paper.”

At rest with respect to red coordinate system (“Earth”).

Moving at constant v as measured in red (“Earth”) coordinate system.

Light co

ne

The proper time in any frame of reference is:

t

x

Δτ = 4 Δτ = 4Moving with –v as measured in blue (“ship”) coordinate system.

At rest in blue (ship) coordinate system.

Light co

ne

The proper time in any frame of reference is:

y

x

Spatial vectors lie on a circle, and have the same length regardless of rotation angle (but different coordinates).

The “frame of reference” where Δx = 0 is the “rest” frame.

We can always make another vector vertical (“at rest”) by rotating the coordinate system by the corresponding angle.

t

x

Spacetime vectors (4-vecs) lie on a hyperbola and have the same “length”(proper time) regardless of velocity (but but different coordinates).

The frame of reference where Δx = 0 is the rest frame.

We can transform to the rest frame of another vector by Lorentz transforming to the corresponding velocity.

t

x

Δτ = 4

Δτ =

4 Δτ = 4

“Take a picture” = Line of constant t

The moving clocks (blue and green) appear to run slowly as seen by the red (rest) observer.

Time Dilation!

Time Dilation

Elapsed time in rest frame (Earth clock)

Speed of moving frame (ship) as seen in rest frame

Elapsed time (ship clock) in moving frame as seen from rest frame

Spacetime Diagrams

t

x

Not moving (at rest) with respect to rest frame (“Earth”)

Moving at velocity v (“ship”) with respect to rest frame (“Earth”)

All x at specific t (photograph!)

Light cone represented by45º lines (x=t) in all reference frames

x

Mirror

T

D

t

x

We define distance by the round-trip travel-time of a pulse of light (“radar method”)

D =½ T

Sends light signal

If light signal goes out some fixed distance in a timet and reflects, then it will come back in thesame amount of time.

-3t

-2t

-t

+3t

+2t

+t

t

x

The line joining the reflection points defines the spatial axis in the rest frame!

Moving Reference Frames

t

x

t

Line which represents somethingmoving at velocity v (spaceship asseen from the Earth) is equivalentto the time axis of the observer insidethe spaceship!!

Sends light signal

If light signal goes out a distance D in a timet and reflects, then it will come back in thesame amount of time according to the moving frame (ship)!-3t

-2t

-t

+3t

+2t

+t

t

x

t

x

The line joining the reflection points defines the spatial axis in the moving reference frame!

t

x

t

x

“Photographs” in rest frame(events in all space at fixed time)

t1

t2

t3

t4

Lines of constant t are parallel to x-axis!

t

x

t

x

“Photographs” in moving frame

t1t2

t3

t4

Lines of constant t are parallel to x-axis!

Proper Time Calibration

t

x

t

x

c

t1 = 1

t2 = 2

t3 = 3

t4 =4

t1 = 1t2 = 2

t3 = 3t4 =4

t

x

t

x

c

t1t2

t3

t4

At time t (parallel to x-axis)… … t < t !!! (clocks on ship run slow)

In rest frame of Earth (t)…

t

x

t

x

c

t1t2

t3

t4

At time t (parallel to x axis)… t < t !! (clocks on Earth appear slow)

In rest frame of ship (t’)…