# Rotations in Space-Time

Building on rotations using geometric algebra, in this section we define a two-dimensional space-time multivector and then investigate a rotation in space as well as a rotation in space-time, the result of which is equivalent to the Lorentz transformation.

• A Vector in Spacetime
• So far we have considered operations on vectors only - linear combinations of the two basis vectors, $$\mathbf{x}$$ and $$\mathbf{y}$$. Here we will consider a more general combination of a vector together with an $$\mathbf{xy}$$ component to form a "multivector". Furthermore, we will see that this extra component behaves geometrically as we expect time to behave in the space-time of Special Relativity and, by extension, in our every day experience. We will then see that we can rotate these space-time multivectors in space without affecting the temporal component, corresponding to changing your frame of reference within a single inertial frame.

• The Lorentz Transformation
• In the previous section we saw that we can rotate a space-time multivector $$\v{A}$$ in space - using a rotation from one normal vector $$\v{b}$$ to another normal vector $$\v{c}$$ to form the rotation $$\v{bc} = e^{\v{xy}\frac{\theta}{2}}$$. We also saw that we can define a normal space-time multivector as $$\v{B} = \v{b}e^{\v{bt}\phi}$$, where $$\v{b}^2 = 1$$. In this section we will consider a space-time multivector $$\v{A}$$ being rotated in space-time - using a rotation from one normal multivector $$\v{B}$$ to another normal multivector $$\v{C}$$, and see that the result is the standard Lorentz transformation.