Projecting spacetime directions
Lowering dimensions
To visualize the geometric elements of an (1,N) dimensional spacetime, we
make a projection into a corresponding N space, thus lowering one
dimension. In the projected space, we can only observe directions, not
magnitudes like lengths or areas.
The set of all vectors which are parallel to a given one (vector direction)
will be represented by a point, a bivector direction by a line, and a trivector
direction by a surface.
Figure 1.1.1: One-dimensional projection
Dynamic app 1.1.1: One-dimensional projection of spacetime
Explanations for the previous (one-dimensional) dynamic app 1.1.1
Drag the green point P to change the direction (red line) which is projected on it.
The event E lies on a unit circumference in spacetime which is centred on the original event O.
The tangent to E on the circumference intersects the horizontal projection line on the point P, which represents all the vectors parallel to E (red line).
The points LF and LP are the projections of the future and past light cones (dashed lines), respectively.
The segment between them (Present) represents all the spacelike directions, which are causally disconnected from the original event O.
The interior of the future light cone (Ft) is shown in blue, and the interior of the Past light cone (Pt) in violet color. Both correspond to timelike directions, which are causally connected to O.
Their projections, the Future and the Past segments, show the same colors.
Figure 1.1.2: Two-dimensional projection
Dynamic app 1.1.2: Two-dimensional projection of spacetime
Explanations for the previous (two-dimensional) dynamic app 1.1.2
Drag the green point P to change the direction (red line) which is projected on it.
The event E lies on the surface of a unit sphere in spacetime which is centred on the original event O.
The tangent to E on the circumference intersects the horizontal projection line on the point P, which represents all the vectors parallel to E (red line).
The Future light cone is shown in blue, and the Past light cone in violet color. Both correspond to timelike directions, which are causally connected to O.
Their projections on the horizontal plane are shown with the same colors.
Figure 1.1.3: Three-dimensional projection
Directions in the projection space
The projection space keeps its directions parallel to the spacelike directions from the original spacetime.
The distance from the centre is a measure of the !amount of time" in the original spacetime direction. A null radius corresponds to pure time, and all the points in the inside of the blue ball correspond to timelike directions.
The points on the blue circle have the same !amount" of time and space, and thus they are purely lightlike directions.
Farther away from the red ball the points correspond to spacelike directions, which on the unit ball get purely spatial (with no time component at all).
The violet ball corresponds to the past lightcone.