Scalar (relativity)

In relativity, a scalar is a number that is the same in all reference frames. In particular, the inner product of two vectors is a scalar.

This implies several things about how we represent vectors in relativity. In particular, our usual idea of a dot product will not work. Instead, we will require an inner product to be between an up and a down vector, which are related by the metric of spacetime.

Given an up vector $a^i$, we write the down vector $a_i$ as

This particularly simple form works only in special relativity in cartesian coordinates.

The negative sign allows us to say that all inner products are scalars. For example, the vector magnitude $a_i a^i$ is a scalar, and therefore an invariant.

Invariant interval

The distance between two events $A$ and $B$ in sapcetime, defined as $\Delta s^2 = |A - B|^2$, is called the invariant interval. Being a vector magnitude, it is invariant, and it’s value tells us what kind of separation the two events have.

If $\Delta s^2 < 0$, the $-c^2 \Delta t^2$ term dominates, and the events have a “time like” separation. There exists a reference frame in which $A,B$ occur at the same point. In this case a signal with speed $v < c$ can connect them.

If $\Delta s^2 > 0$, the distance in space dominates and the events are “space like” separated. There exists a reference frame in which $A,B$ are simultaneous. $\Delta s = \sqrt{\Delta s^2}$ is the “proper separation” in that frame. There also exist frames where $A,B$ happen in different orders.

If $\Delta s^2 = 0$, the space and time differences cancel and the events are “light like” separated. Since $c \Delta t = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$, the events can be connected by a light pulse.

The graphic below shows the characterization of the invariant interval between the event at the middle of the light cone and the other points.