*83**Advanced Space Propulsion Based on Vacuum (Spacetime Metric) Engineering*The appropriate mathematical evaluation tool is use of the

*metric tensor* that describes the measurement of spacetime intervals. Such an approach, well-known from studies in GR (general relativity) has the advantage of being model-independent, i.e.,

does not depend on knowledge of the specific mechanisms or dynamics that result in spacetime alterations, only that a technology exists that can control and manipulate (i.e., engineer) the spacetime metric to advantage. Before discussing the predicted characteristics of such engineered spacetimes a brief mathematical digression is in order for those interested in the mathematical structure behind the discussion to follow.

As a brief introduction, the expression for

the dimensional line element *ds*2

in terms of the metric tensor

*g*µ

*v*is given by

2

*ds**g**dx dx*µ

ν

µν

=

(1)

where summation over repeated indices is assumed unless otherwise indicated. In ordinary Minkowski flat spacetime a (4- dimensional) infinitesimal interval

*ds* is given by the expression (in Cartesian coordinates 2

2 2

2 2

(

)

*ds**c dt**dx**dy**dz*=

−

+

+

(2)

where we make the identification

*dx*0

=

*cdt*,

*dx*1

=

*dx*,

*dx*2

=

*dy*,

*dx*3

=

*dz*, with metric tensor coefficients

*g*00

= 1,

*g*11

=

*g*22

=

*g*33

= -1,

*g*µ

*v** *= 0 for

µ ≠

*v*.

For spherical coordinates in ordinary

Minkowski flat spacetime2 2

2 2

2 2

2 2

2

sin

*ds**c dt**dr**r d**r**d*θ

θ where

*dx*0

=

*cdt*,

*dx*1

=

*dr*,

*dx*2

=

*d*θ,

*dx*3

=

*d*ϕ

*, *with metric tensor coefficients

*g*00

= 1,

*g*11

= -1,

*g*22

= -

*r*2

,

*g*33

= -

*r*2

sin

2

θ,

*g*µ

*v*= 0 for ≠

*v*.

As an example

of spacetime alteration, in a spacetime altered by the presence of a spherical mass distribution

*m* at the origin (Schwarzschild-type solution) the above can be transformed into [10]

(

) (

)

1 2

2 2

2 2

2 2

2 2

2 2

2 2

1 1

1 1

1

sin

*Gm rc**Gm rc**ds**c dt**dr**Gm rc**Gm rc**Gm rc rd iidi ϕ*

−

−

−

=

−

+

+

− +with the metric tensor coefficients *g*

µ

*v*

modifying the Minkowski flat-spacetime intervals *dt*, *dr*, etc, accordingly.

As another example of spacetime alteration, in a spacetime altered by the presence of a *charged* spherical mass distribution

(*Q*, *m*) at the origin (Reissner-Nordstrom-type solution) the above can be transformed into [11]

(

)

(

)

(

) (

)

2 4

2 2

2 2

0 2

2 2

2 1

2 4

2 2

0 2

2 2

2 2

2 2

2 2

2 4

1 1

1 4

1 1

1 1

sin

*Q G*

*c*

*Gm rc*

*ds*

*c dt*

*Gm rc*

*r*

*Gm rc*

*Q G*

*c*

*Gm rc*

*dr*

*Gm rc*

*r*

*Gm rc*

*Gm rc*

*r*

*d*

*d*

πε

πε

θ

θ ϕ

−

−

=

+

+

+

−

−

+

+

+

− +with the metric tensor coefficients *g*

µ

*v*

again changed accordingly. In passing, one can note that the effect on the metric due to charge *Q* differs in sign from that due to mass m, leading to what in the literature has been referred to as *electrogravitic*

*repulsion* Similar relatively simple solutions exist fora spinning mass

(Kerr solution, and fora spinning electrically charged mass

(Kerr-Newman solution. In the general case, appropriate solutions for the metric tensor can be generated for arbitrarily- engineered spacetimes, characterized by an appropriate set of spacetime variables *dx*

µ

and metric tensor coefficients *g*

µ

*v*

*.* Of significance now is to identify the associated physical effects and to develop a Table of such effects for quick reference.

The first step is to simply catalog metric effects, i.e., physical effects associated with alteration of spacetime variables,

and save for Section 4 the significance of such effects within the context of advanced aerospace craft technologies.

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