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

• Einstein discovered that time and distance changes when moving from one inertial reference frame to another when travelling at speeds close to the speed of light
  • In other words, these reference frames are not absolute

• However, some quantities are the same in all inertial frames. These are called invariant
• These are:
  • Proper time, t₀
  • Proper length, L₀
  • Space-time interval, Δs

• These are a product of Einstein's second postulate
• In Galilean relativity:
  • Space and time are the same in all reference frames, i.e. and 

• In special relativity:
  • These are replaced with a space-time interval, as space and time are connected together as 4 coordinates (x, y, z, t) for an event
  • Motion can be represented as spanning both space and time using this coordinate system

• The diagram below shows a person moving in both space x and z and in time t
  • They can also move in the y direction, but 4 dimensions are not possible to draw accurately here (in 3-dimensional space)

Motion in space-time. The length of the arrow for the space-time interval is the same for all inertial reference frames

• An interval in space-time is an invariant quantity in all inertial reference frames and is defined as:

• Where:
  • = time interval / separation (s)
  • = the speed of light
  • = spacial separation (m)
  • = space-time interval (m)
• This means that in two inertial reference frames, although and  will be different in both frames,  will be the same
• These will be used in space-time diagrams

• In special relativity, we have found that distances and times are relative
• This means the length of an object and the time interval of an event changes when observed in frames that are moving relative to that object
  • More on this is explored in length contraction and time dilation
• We need to define the difference mathematically between the time and lengths measured between each frame, to know which one is being referred to

What is Proper Time and Proper Length?

• Proper time interval, Δt₀ is defined as:

The time interval between two events measured from within the reference frame in which the two events occur at the same place

• Proper length, L₀ is defined as:

The length measured in a reference frame where the object is at rest (relative to the observer)

• These can be measured either in the moving frame S' or a rest frame S
  • This depends on the reference frame you are calculating the length and time from

• For example, if a person in moving frame S' (e.g. on a train) measures the length of a book, they are at rest relative to the book
  • They measure the distance between points x₂' and x₁'
  • This is the proper length, L₀ although they are technically moving (but they don't know this - otherwise it would go against Einstein's first postulate)

• However, for an observer in frame S, at rest (e.g. on a platform
  • They will measure the distance between points x₂ and x₁
  • This a shortened length, L

The observer in S' is measuring the proper length, because they are at rest relative to the object

• For example, if a person in the stationary frame S (e.g. on the platform) measures the length of a book, they are now at rest relative to the book
  • They measure the distance between points x₂ and x₁
  • This is the proper length, L₀ 

• However, an observer in frame S', which is moving (e.g. on a train) sees themselves at rest but instead sees frame S as moving (remember, motion is relative)
  • They will measure the distance between points x₂' and x₁'
  • This a shortened length, L

The observer in S is measuring the proper length because they are at rest relative to the object

• The same rules apply for the proper time, t₀, except that the time measured from a frame moving relative to an event will be measured longer