metricGet_VanDenBroeck

Description

Create the Van Den Broeck warp solution in a defined spacetime grid.

Van Den Broeck Metric

The Van Den Broeck solution modifies an Alcubierre geometry in the following way:

$ds^2 = -dt^2+ B^2(r_s)[(dx^2 - v_s(t) f(r_s)^2 \ dt)^2 + dy^2 + dz^2]$

where $r_s$ is:

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

The added function $B(r_s)$is twice differentiable such that:

$B(r_s) = 1+ \alpha \ \ \text{for} \ \ r_s < R$

$1 < B(r_s) \leq 1 + \alpha \ \ \text{for} \ \ \tilde{R} \leq r_s < \tilde{R} + \tilde{\Delta}$

$B(r_s) = 1 \ \ \text{for} \ \ \tilde{R} + \tilde{\Delta} \leq r_s$

where $\alpha$is a constant and $1 + \alpha$ is the factor of spatial expansion. The function of $f(x)$must be selected to have the following constraints:

$f(r_s) = 1 \ \ \text{for} \ \ r_s < R$

$0 < f(r_s) \leq 1 \ \ \text{for} \ \ \tilde{R} \leq r_s < \tilde{R} + \tilde{\Delta}$

$f(r_s) = 0 \ \ \text{for} \ \ \tilde{R} + \tilde{\Delta} \leq r_s$

For more details on the Van Den Broeck metric, please read:

A `warp drive' with more reasonable total energy requirementsarXiv.org

Method

The metric is constructed using the parameters of the Van Den Broeck metric in the user-defined spacetime grid parameters. In this setup R1 and sigma1 repersent $\tilde{R}$ and $\tilde{\Delta}$ . R2 and sigma2 repersent $R$ and $\Delta$.

The comoving version of this metric called metricGet_VanDenBroeckComoving 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_VanDenBroeck(gridSize,worldCenter,v,R1, sigma1 ,R2, sigma2 ,alpha, gridScale)

[metric] = metricGet_VanDenBroeckComoving(gridSize,worldCenter,v,R1, sigma1 ,R2, sigma2 ,alpha, 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.
R1	1x1 array	double	Radius of the spatial expansion.
sigma1	1x1 array	double	Thickness factor of the spatial expansion. Note this uses the Alcubierre shape function.
R2	1x1 array	double	Radius of the shift vector.
sigma2	1x1 array	double	Thickness factor of the shift vector. Note this uses the Alcubierre shape function.
alpha	1x1 array	double	Spatial expansion factor
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	Van Den Broeck solution returned as the metric tensor object.