S18. Spacetime geometry

The purpose of notes on the 3d geometry was to show that length of the vector was invariant even though we make various complex transformations of the coordinate axes. The question we want to address here is that is this true for the spacetime geometry too?

What we do now is to make use of the idea of 3d geometry to introduce the idea of a vector describing the separation of two events occuring in spacetime. The essential idea will be to show that the coordinate of an event have the transformation analogous to the rotational transformation which we noted in the previous section for ordinary three vectors, though with some surprising differences. Lets us consider two events $E_1$ with coordinates $(x_0,x_1,x_2,x_3)$ in the frame $S$ and $E_2$ with coordinates $(x'_0,x'_1,x'_2,x'_3)$ in $S'$,

For event $E_1$ and $E_2$we have the following lorentz transformation,


Then we can write event separation transformation as;

or in more visual ways as,

Now, as we can see that the form of the transformation looks like what we have seen in the rotational transformation in the previous section. Therefore, at first it is tempting to interpret this equation as relating the two components as having properties analogous to the length and angle between vectors  for ordinary three vectors which are independent of the choice of reference frame. However, it turns out that that the length that is invariant in for different reference frame for this geometry is the quantity,

 $\Delta s^2=-(c\Delta t)^2+(\Delta x)^2+(\Delta y)^2+(\Delta z)^2$

The invariant quantity $\Delta s^2$ is called the interval between the two events $E_1$ and $E_1$. We can compare this interval as being analogous to the length but they are fundamentally different quantities. Interval $\Delta s$ can be positive, negative or zero. The angle between two such intervals is also the same in all the reference frame.

i.e.$-(c\Delta t_1)(c\Delta t_2)+\Delta x_1\Delta x_2+\Delta y_1\Delta y_2+\Delta z_1\Delta z_2$

has the same value in different reference frame.

The quantity $\Delta \vec{s}$ is then understood to correspond to a property of spacetime representing the separation between two events which has an absolute existence independent of the choice of reference frame and is known as a four-vector.
In the literature of relativity, space-time coordinates and the energy/momentum of a particle are often expressed in four-vector form. They are defined so that the length of a four-vector is invariant under a coordinate transformation. This invariance is associated with physical ideas. The invariance of the space-time four-vector is associated with the fact that the speed of light is a constant. The invariance of the energy-momentum four-vector is associated with the fact that the rest mass of a particle is invariant under coordinate transformations. 

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