# 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.

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 [d
^{2}=x^{2}+y^{2}+z^{2}], i.e. this aspect does conform to Pythagoras’ theorem.

- However, the
*‘spacetime interval [s]’ is a*4-dimensional measure that is defined by [s^{2}=t^{2}-d^{2}], 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 45^{o
}to the origin.

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] c^{2} *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]

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]

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].