Concepts in Spacetime

Having established the basics of spacetime in previous discussions, we can now turn our attention to some of the concepts used describe spacetime. First, we possibly need to reflect on the fact that spacetime is a 4-dimensional union of the classical concept of 3-dimensional space plus absolute time, as inferred by the Galilean transforms.

So what new concepts emerge from the union of space and time into a single 4-dimensional continuum?

Possibly, the key issue to consider is the concept of the ‘spacetime interval’, which we shall initially, and possibly misleading, describe as the ‘distance’ between two events in 4-dimensional spacetime. We might also try to visualise this ‘distance’ in terms of the diagram below, which plots ‘time interval [t]’ on the vertical axis and ‘spatial distance [d]’ on the horizontal axis, where the different units of [t] and [d] are unified via the relationship [d=ct] with [c] being the speed of light.

1

So, with reference to the diagram above, event [A] at the origin of the axes is separated in spacetime from events [B] and [C]. Just by way of general observation, we may note that the time component of the ‘distance’ between [A-B] is greater than its spatial component. In contrast, the spatial component of the ‘distance’ between [A-C] seems to be greater than its time component. However, in contradiction to what the diagram might suggest, the axial components of time [t] and space [d] do not combine in some form of vector addition analogous to Pythagoras’ theorem. Therefore, the following bullets will try to clarify some of the basic rules that do apply:

  • The spatial distances [d] is still constructed from its 3-dimensional [xyz] components via [d2=x2+y2+z2], i.e. this aspect does conform to Pythagoras’ theorem.

  • However, the ‘spacetime interval [s]’ is a 4-dimensional measure that is defined by [s2=t2-d2], where the square of the components [t] and [d] are subtracted, not summed.

In essence, the diagram above is a simplistic example of a 2-dimensional spacetime diagram, which features the ‘light-cone’ formed when plotting [c] in terms of a series of time [t] versus distance [d] values. While it is not possible to fully illustrate 4-dimensional spacetime, we can extend the previous diagram by extending the representation of space along the x-axis to the xz-plane, while still retaining time along the vertical axis. In the 3-dimensional spacetime diagram below, the path of 3 chronological events, i.e. [-A, A, +A] are shown, where [-A] represents some event in the past, which is able to affect [A] because it originates within the past light cone. In a similar fashion, [A] is able to affect [+A] in the future, because [+A] is inside the light cone of [A]. Typically, the speed of light is normalised to [c=1], which then allows an offset in space to be equated to an offset in time, where [c] acts as a conversion factor between the units of distance and the units of time. On this basis, anything travelling at the speed of light [c] moves along the surface of the light cone at an angle 45o to the origin.  

  2

However, the purpose of the diagram is to try to further illustrate some of the spacetime concepts at work. By way of a reference, let us assume that the red-orange dots, in the diagram above, corresponds to a clock travelling a distance of 3 light-years in 5 years at a constant velocity [v=0.6]. Relativity tells us that the time on the moving clock [+A] must be slower than a stationary clock that remains at [A]. However, if we generalise this statement, it means that any path through spacetime must conceptually carry its own clock, which measures the `proper time [τ]` in the frame of motion. The proper time [τ] on our moving clock can also be calculated using the following equation:

  [1]      c2 τ 2   = c 2 t 2  – (x 2  + y 2  + z 2 )

However, [1] can be simplified even further in the context of the previous 2-D spacetime diagram by restricting the spatial dimensions to just [x] and normalising [c=1]: 

  [2]      τ = √(t 2  – x 2 )

While [2] is easier to work with, especially when trying to draw diagrams restricted to two-dimensions, i.e. [x] and [t], it is still equivalent to the Lorentz transform shown below, although this is not always immediately obvious: 

  [3]        3

Rather than going through the mathematical derivation to prove the equivalence of [2] and [3], we might simply plug in the figures from the diagram into both equations and compare the results. Let us start with [2]:

  [4]      τ = t' = √(t 2  – x 2 ) = √(5 2  – 3 2 ) = √(25 – 9) = √16 = 4 years

When we turn our attention to [3], we immediately realise that we need to know the velocity [v]. However, in this example,  the velocity [v=0.6c] and so we might realise that by plotting the values of [t] and [x] on the spacetime diagram allows a line to be drawn from the origin, which creates an angle with the axis. By definition, the slope of this line corresponds to the velocity [v=x/t], which is also reflected in the ratio [v/c]. As such, it points to the geometric solution of [2], implicit in [3]:

  [5]        5

Therefore, [2] and [3] both determine the proper time [t], i.e. the time on our moving clock [+A], which would register 4 years as opposed to the 5 years on the stationary clock at [A].