Space-Time Diagrams (HL) (HL IB Physics)

• Spacetime (or Minkowski) diagrams represent an object's motion in spacetime
• They help to visualise
  • Time dilation
  • Length contraction
  • Simultaneity

• Since 4D (x, y, z, t) diagrams cannot be drawn on a 2D page, we collapse 3D space (x, y, z) into 1 spacial dimension and keep time as its own dimension
• This gives a spacetime diagram
  • Lines drawn on a spacetime diagram are called world lines
• Instead of the usual distance-time graphs, we plot
  • The horiztonal axis as x
  • The vertical axis as ct

Objects moving slower are represented by a steeper gradient on a spacetime gradient

• Note that both axes have dimensions of length
  • This makes it easy to compare values on one axis and another
• This means the worldlines have a gradient of
  • where the velocity is
• This means:
  • The steeper the gradient, the slower the object is moving
  • The shallower the gradient, the faster the object is moving
• ct was first seen when the spacetime interval was introduced, and c is chosen deliberately so our diagram is oriented around the speed of light
  • ct is a sort of 'distance in time'

Worldline at Rest

A worldline for an object at rest

• P is an object at rest
  • It has an infinite slope meaning v = 0
• It is only moving in time, not in space

Worldline at Constant Velocity

A worldline at constant velocity has to have a gradient steeper than 1

• Q is an object at constant velocity
• The object moves some distance over some time
• For an object moving at the speed of light, c (e.g. a massless photon), its worldline has a gradient of 1 (a 45^o angle)
• Since nothing can travel faster than the speed of light (i.e. v cannot be greater than c), then objects can only have a gradient greater than 1
  
  • Therefore, a gradient of 1 is the lowest possible gradient
• Q's motion does not need to start at the origin. As long as it has a gradient of less than 1, it can start anywhere on the x-axis

Worldline accelerating

All points on a worldline representing an object accelerating must have a gradient steeper than 1

• R is an object with a varying velocity that represents possible motion
  • It has a small gradient (larger velocity), which increases (decreasing velocity)
  • Its gradient never gets less than 1
• S is an object with a varying velocity that represents impossible motion
  • At one point, it has a gradient of less than 1 implying a velocity greater than c
  • Even though it does not physically cross the v = c gradient line, it is still not possible because a portion of the line has a gradient of less than one

Multiple Reference Frames

• Every point on a spacetime diagram represents an event, for example, an object moving
• More than one inertial reference frame can be represented on a spacetime diagram
  • ct and x represent the co-ordinate axes for an observer in frame S
  • ct' and x' represent the co-ordinate axes for an observer in frame S' (moving at speed v with respect to frame S)
• Therefore, we can combine two separate spacetime diagrams for different inertial reference frames moving at constant speed relative to each other
  • The axes for the ct' and x' are at an angle
• The worldline T shows the equivalent worldline of P, but now in the S' reference frame
  • This represents an object at rest in its own co-ordinate system

Worldline of a particle at rest in the reference frame S'

• The axes are tilted because the ct' reference frame is travelling at speed v relative to the ct reference frame
  • The x' axis must also be tilted in order for the speed of light (the dashed line in the middle) to be the same in both reference frames
• The scales on the time axes ct and ct′ and on the space axes x and x′ of two inertial reference frames moving relative to one another are not the same and are defined by lines of constant spacetime interval
• If an event occurs (such as a flash of light), both reference frames will measure a different time and different position with respect to each other
  • This can be seen on a spacetime diagram

An event is shown in reference frames S and S' with differing values of distance and time

• The event in the S reference frame occurs at (X, cT)
• The event in the S' reference frame occurs at (X', cT')
• The co-ordinates in the S' reference frame are determined by lines 1 and 2
  • Line 1 is a line parallel to the x' axis
  • Line 2 is a line parallel to the ct' axis
• The clocks in both frames show zero at the origins where two frames collide i.e. both observers start their clocks at the same time to measure any time intervals

Simultaneity

• We can now see that simultaneous events in one frame are not simultaneous in another moving inertial reference frame
• Let's go back to Observers A and B in Simultaneity in Special Relativity
  • We can see that Observer B sees the light reach points X and Y at the same time, whilst Observer A (in the ct'–x' co-ordinate system) sees the light from the lamp reach point X before point Y on a spacetime diagram

Simultaneity is not possible for two reference frames moving relative to each other

Time Dilation

• Consider two flashes of light at x = 0 in the S reference frame that occur one after another
• When these flashes are observed in the S' frame, we can see the time between the flashes is longer
  • The time between them has increased (dilated)

Spacetime diagrams representing time dilation

• Another difference is that in the S' reference frame, the first flash now occurs on the -x axis
  • This just means it takes place to the left of the observer

Length Contraction

• Consider a rod measured in the S reference frame where the rod moving relative to S
• The rod has the same speed as it does in the S' reference frame (which is also moving relative to S)

Spacetime diagrams representing length contraction

• The length of the rod is measured by measuring each side at the same time
  • The observer in frame S will measure a length L
• When the observer in frame S' measures the rod, they see the rod as stationary (as it is moving at the same speed as the observer)
  • The observer in frame S will measure a length L'
• L is shorter than L', which means that the length has been shortened (contracted) when measured by Observer S, who is moving relative to the rod
  • Although it is the rod that is moving, remember, it is at rest in its own reference frame and Observer S is moving relative to it
• This occurs from the fact that measurements that are simultaneous in one reference frame are not simultaneous in another

• The worldline for a moving particle on a spacetime diagram using the x-ct axis is a diagonal line

• The velocity of the particle can be calculated by the angle of the moving particle's worldline with the ct axis
• When we are using the x-ct axis, we can see that:

• From mechanics, we know that velocity is the rate of change of displacement:

• Therefore:

• Where:
  • θ = angle between the world line and the ct axis (°)
  • v = velocity of the object (m s^–1)
  • c = speed of light