metricGet_Lentz

Lentz warp solution implementation

Description

Create a version of the Lentz warp solution in a defined spacetime grid.

Lentz Metric

The Lentz solution takes a soliton approach which solves for a potential that satisfies a linear wave equation in spherical coordinates:

$\partial_x^2 \phi + \partial_y^2 \phi - \frac{2}{v^2_h} \partial_z^2 \phi = \rho$

where the shift vector $\beta$ is given in two dimensions where:

$\beta_i = \partial_i \phi$

The specific solution that Lentz uses has the shift vectors in $x$ and $z$ found in Green's form:

$\beta_z = \frac{1}{4v_h}\int dx^\prime \rho \left( z - \frac{|\Delta x|}{v_h}, |x^\prime|+|y| \right)$

$\beta_x = \frac{1}{4v_h^2}\int dx^\prime \text{sign}(\Delta x) \rho \left( z - \frac{|\Delta x|}{v_h}, |x^\prime|+|y| \right)$

For more details on the Lentz metric, please read:

Breaking the Warp Barrier: Hyper-Fast Solitons in...arXiv.org

Method

The metric is constructed using a representation of the Lentz metric as described in the first publication. An internal function in the main metric constructor called getWarpFactorByRegion is called which returns a map of the shift vector values as used in Figure 2 of https://arxiv.org/abs/2006.07125. This function scales the shape from the Lentz solution using the scale parameter.

The comoving version of this metric called metricGet_LentzComoving 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_Lentz(gridSize,worldCenter,v,scale, gridScale)

[metric] = metricGet_LentzComoving(gridSize,worldCenter,v,scale, 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 the Lentz shape template 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.
scale	1x1 array	double	Scaling parameter for the Lentz solution template. Default value is largest gridSize/7.
gridScale	1x4 double	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	Lentz solution returned as the metric tensor object.