Alcubierre Drive - Mathematics of The Alcubierre Drive

Mathematics of The Alcubierre Drive

Using the ADM formalism of general relativity, the spacetime is described by a foliation of space-like hypersurfaces of constant coordinate time t. The general form of the metric described within the context of this formalism is:

where

is the lapse function that gives the interval of proper time between nearby hypersurfaces,

is the shift vector that relates the spatial coordinate systems on different hypersurfaces

is a positive definite metric on each of the hypersurfaces.

The particular form that Alcubierre studied is defined by:

where

and

with and arbitrary parameters. Alcubierre's specific form of the metric can thus be written;

With this particular form of the metric, it can be shown that the energy density measured by observers whose 4-velocity is normal to the hypersurfaces is given by

where is the determinant of the metric tensor.

Thus, as the energy density is negative, one needs exotic matter to travel faster than the speed of light. The existence of exotic matter is not theoretically ruled out; however, generating enough exotic matter and sustaining it to perform feats such as faster-than-light travel (and also to keep open the 'throat' of a wormhole) is thought to be impractical. Low has argued that within the context of general relativity, it is impossible to construct a warp drive in the absence of exotic matter.