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Building Spacetime Diagrams. PHYS 206 – Spring 2014. The “proper length” (radius) of the circle in any rotated frame of reference is:. t. But their coordinates do not agree!. Δr = 4. Δr = 4. Δr = 4. x. Think of the red line as the “rest” frame (inertial clock with v=0). . y.

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### Building Spacetime Diagrams

PHYS 206 – Spring 2014

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

t

But their coordinates do not agree!

Δr = 4

Δr = 4

Δr = 4

x

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

y

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

Δr = 4

x

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

y

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

Δr = 4

Δr = 4

x

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

y

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

Δr = 4

Δr = 4

x

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

The proper time in any frame of reference is:

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

Light cone

Δτ = 4

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

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

Δτ = 4

x

t

The proper time in any frame of reference is:

Light cone

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

Δτ = 4

At rest in blue (ship) coordinate system.

Δτ = 4

x

y

t

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.

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.

x

t

“Take a picture” = Line of constant t

Δτ = 4

Δτ = 4

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

Δτ = 4

x

Time Dilation

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

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

Elapsed time in rest frame (Earth clock)

t

x

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

All x at specific t (photograph!)

Light cone represented by

45º lines (x=t) in all reference frames

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

t

x

We define distance by the round-trip

travel-time of a pulse of light (“radar method”)

D =½ T

T

Mirror

D

x

t

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

+3t

+2t

+t

x

-t

-2t

If light signal goes out some fixed distance in a time

t and reflects, then it will come back in the

same amount of time.

Sends light signal

-3t

t

x

t

Line which represents something

moving at velocity v (spaceship as

seen from the Earth) is equivalent

to the time axis of the observer inside

the spaceship!!

t

t

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

+3t

x

+2t

+t

x

-t

If light signal goes out a distance D in a time

t and reflects, then it will come back in the

same amount of time according to the moving frame (ship)!

-2t

-3t

Sends light signal

t

t

x

x

“Photographs” in rest frame

(events in all space at fixed time)

t4

t3

t2

t1

Lines of constant t are parallel to x-axis!

t

t

x

x

“Photographs” in moving frame

t4

t3

t2

t1

Lines of constant tare parallel to x-axis!

t

t

c

x

x

t4 =4

t3 = 3

t4 =4

t3 = 3

t2 = 2

t2 = 2

t1 = 1

t1 = 1

In rest frame of Earth (t)…

t

t

c

x

x

t4

t3

t2

t1

At time t (parallel to x-axis)…

… t< t !!! (clocks on ship run slow)

In rest frame of ship (t’)…

t

t

c

x

x

t4

t3

t2

t1

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