metricGet_Alcubierre

Standard Alcubierre metric setup

Description

Create the Alcubierre warp solution in a defined spacetime grid.

Alcubierre Metric

The Alcubierre metric uses only a single shift vector which is defined by a shape function $f(r_s)$:

$f(r_s) = \frac{\tanh(\sigma(r_s + R)) - \tanh(\sigma(r_s - R))}{2 \tanh (\sigma R)}$

where $r_s$is the coordinate center defined as:

$r_s = [(x-x_s(t))^2 + y^2 + z^2]^{1/2}$

where $x_s(t)$ moves through spacetime at a rate of $v$ along $x$. The shift vector $\beta_1$ is then defined using the shape function:

$\beta_1 = v \ f(r_s)$

The metric is then given (in the line element representation) by:

$ds^2 = -1+v^2 f(r_s)^2 \ dt^2 + 2 v f(r_s) \ dx dt + dx^2 + dy^2 + dz^2$

For more details on the Alcubierre metric, please read:

The warp drive: hyper-fast travel within general relativityarXiv.org

Method

The metric is constructed using the parameters of the Alcubierre metric in the user-defined spacetime grid parameters.

The comoving version of this metric called metricGet_AlcubierreComoving has the same inputs but requires that the gridSize along t = 1 and will return the metric in the Galilean comoving frame.

Syntax

[metric] = metricGet_Alcubierre(gridSize,worldCenter,v,R,sigma, gridScale)

[metric] = metricGet_AlcubierreComoving(gridSize,worldCenter,v,R,sigma, gridScale)

Input Arguments

blue are required inputs.

orange are optional inputs with native default values.

Inputs	Format	Type	Description
gridSize	1x4 array	integer	The size of the world specified as: $[t, x, y, z]$
worldCenter	1x4 array	double	The center of the world, which defines the center of $r_s$ as a 4-vector, specified as: $[t,x,y,z]$
v	1x1 array	double	Speed of the warp drive, given as a factor of c.
R	1x1 array	double	Radius of the warp bubble.
sigma	1x1 array	double	Width parameter of the warp bubble.
gridScale	1x4 array	double	Unit scaling factor of the grid dimensions defined relative to gridSize. This determines the resolution of the grid along each dimension. Specified as: $[t_{scale}, x_{scale}, y_{scale}, z_{scale}]$ The default value is [1, 1, 1, 1].

Output Arguments

Outputs	Format	Type	Description
metric	struct	object	Alcubierre solution returned as the metric tensor object.