A T HEORETICAL WARP D RIVE : T HE
M ATHEMATICS OF FASTER T HAN L IGHT
T RAVEL
By
Nikki Holtzer
A SENIOR RESEARCH PAPER PRESENTED TO THE DEPARTMENT OF MATHEMATICS AND
COMPUTER SCIENCE OF STETSON UNIVERSITY IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE
STETSON UNIVERSITY 2013
1
Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
0.1
4
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
4
2 The Schwarzschild Metric
5
3 The Alcubierre Metric
9
4 Conclusion
14
2
List of Figures
1
Geometric interpretation of the four dimensional spherical coordinate system for the Schwarzschild
metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2
Geometric interpretation of the Schwarzschild metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3
Geometric interpretation of the Alcubierre metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4
Computed Geometry of the Alcubierre metric.
5
Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6
Visual Representation of a Light Cone.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3
0.1
Abstract
A Theoretical Warp Drive: The Mathematics of Faster Than Light Travel
By Nikki Holtzer
May 2014
Advisors: Dr. Thomas Vogel
Department: Mathematics and Computer Science
Previously, work by Miguel Acubierre has shown the possibility of a spacecraft travelling faster than the speed of
light. His 1994 paper developed a novel metric motivated by the Einstein field equations. Instead of travelling within
normal space-time, which inhibits an acceleration faster than the speed of light, AcubierreâĂŹs spacecraft would
travel across distances by contracting space in front and dilating the space behind the spacecraft. In theory, this
change in space can be accomplished by placing a spherical object between two regions of space-time creating a warp
âĂIJbubbleâĂİ. Although this metric is mathematically consistent with the field equations it requires a negative
energy density and thus cannot be constructed. Recently, Harold White has attempted to extend AcubierreâĂŹs
research in hopes of creating something physically meaningful. He attempts to perturb the geometry of the warp
bubble (i.e. the object altering space) to minimize the amount of energy and mass required. In this project, we
propose that a utilization of optimization techniques will provide the appropriate geometry of the warp bubble
that effectively minimizes both the energy and mass required. Thus far, we have comprehensively analyzed the
Schwarzschild Metric and the Alcubierre Metric that will provide the basis for our future work.
1
Introduction
The theory of gravitation, developed by Albert Einstein, established the standard principle that the gravitational
attraction between two given masses is a direct result of these supposed masses warping space and time. Einstein’s
1915 publication of General Theory of Relativity gave rise to a set of ten equations, known collectively as the
Einstein Field Equations. These equations ascertain a description of the surrounding gravitational field of a mass.
They illustrate the effects an object has on the curvature of space and how this said curvature contorts the matter
in three spatial directions. This Einstein Field Equations are collectively represented by (1.1).
Rµν =
−8πG
c4
1
Tµν − T gµν
2
4
(1.1)
2
The Schwarzschild Metric
Since his covariant field equations were nonlinear, Einstein assumed that his description of general relativity was
unsolvable. The solutions to the field equations are expressed in terms of a metric. Metrics describe the configuration
of spacetime including the inertial motion of the mass. Thus, to say the field equations were "unsolvable" meant
a metric could not be derived from the field equations. However, approximately a year after Einstein’s paper was
published, Karl Schwarschild published the first exact solution to the field equations aside from the trivial flat space
solution. The Schwarzschild metric is the most general vacuum solution to the Einstein Field Equations that is
spherically symmetric. It demonstrates the gravitational field outside of a spherical, uncharged, non-rotating mass.
The Schwarzschild Metric is given by (2.2).
ds2 = B (r) c2 dt2 − A (r) dr2 − r2 dθ2 − r2 sin2 θdφ2
(2.2)
This metric tensor can be written in matrix form as follows:

gσν

A (r)
0
0
0
0
r2
0
0
0
0
r2 sin2 θ
0
0
0
0
B (r)
1
A(r)
0
0
0
1
r2
0
0
0
0




=











(2.3)
Therefore the inverse metric tensor is as follows:

g σν




=



0
r2
1
sin2 θ
0
0
0
0
1
B(r)









(2.4)
To verify that the Schwarzschild Metric is indeed an exact solution to the Einstein Field Equations we first calculate
all the respective nonzero Christoffel symbols of the second kind corresponding to this metric. The Christoffel symbols
describe parallel transport in curved surfaces or manifolds (n-dimensional geometry of space). Therefore, Christoffel
symbols cannot be calculated for every possible solution to the Einstein Field Equations because parallel transport
arises from the existence of a covariant derivative or connection.
We define the Christoffel Symbols to be the product of the Riemann Tensor and the inverse metric tensor (2.5). Note
5
the subscripts λ and µ denote the respective row and column of the metric tensor in matrix form and the superscript
σ denotes the Einstein summation indice.
Γσλµ
1
= g σν
2
∂gλν
∂gµλ
∂gµν
+
−
λ
µ
∂x
∂x
∂xν
(2.5)
To calculate the Riemann tensor we rewrite Riemann component from above.
Γσλµ =
1
(gσλ,µ + gσµ,λ − gλµ,σ )
2
(2.6)
If we consider the first entry in our metric tensor matrix, which appears in the rth row and the rth column, we can
compute the first element of the Riemann tensor and thus the first of the Christoffel symbols. Consider, Γrrr . This
implies,
Γrrr =
1
A0
1
(∂r(A(r)) + ∂r(A(r)) − ∂r(A(r))) = (∂r(A(r))) =
.
2
2
2
Multiplying, the previous solution by g rr will yield the first Christoffel symbol. Hence,
Γrrr =
1 A0
A0
=
.
A 2
2A
Continuing in a similar fashion will yield nine surviving Christoffel symbols (2.7).
A0
2A
−r
A
Γrrr
=
Γrθθ
=
Γθθr
Γtrt
1
r
−r sin2 θ
=
A
−r sin2 θ
= Γφφr =
A
−B 0
=
2A
B0
= Γttr =
2B
Γθφφ
= − sin θ cos θ
Γφθφ
=
Γrφφ
Γφrφ
Γrtt
=
(2.7)
Γθrθ =
Γφφθ = cot θ
(2.8)
Furthering the verification process of the Schwarzschild metric lies within the development of the Ricci tensor. While
6
the Riemann tensor described the curvature of any given subplane or hypersurface, the Ricci tensor provides an
average of the curvatures of all given sub planes. It is a measure of the degree to which the geometry of space
arising from the Riemann tensor differs from that of ordinary Euclidean geometry. We define the Ricci tensor with
an additional tensor equation (2.9).
ρ
Rµν = Rµρν
=
∂Γρµρ
∂Γρµν
−
+ Γσµρ Γρσν − Γσµν Γρσρ
∂xλ
∂xρ
(2.9)
To solve for the Ricci tensor solutions we first let 0, 1, 2 and 3 denote the coordinates r, θ, φ, and t respectively.
Therefore,
R00
∂
Γ000 + Γ101 + Γ202 + Γ303 − Γ000,0 + Γ100,1 + Γ200,2 + Γ300,3
∂r
=
Γρ0ρ,0 − Γρ00,0 + Γσ0ρ Γρσ0 − Γσ00 Γρσρ =
+
(Γ000 Γ000 + Γ100 Γ010 + Γ200 Γ020 + Γ300 Γ030 + Γ001 Γ100 + Γ101 Γ110 + Γ201 Γ120 + Γ301 Γ130 + Γ002 Γ200 + Γ102 Γ210
+
Γ202 Γ220 + Γ302 Γ230 + Γ003 Γ300 + Γ103 Γ310 + Γ203 Γ320 + Γ303 Γ330 ) − (Γ000 Γ000 + Γ100 Γ010 + Γ200 Γ020 + Γ300 Γ030
+
Γ000 Γ101 + Γ000 Γ202 + Γ000 Γ303 + Γ100 Γ010 + Γ100 Γ111 + Γ100 Γ212 + Γ100 Γ313 + Γ200 Γ020 + Γ200 Γ121 + Γ200 Γ222
Γ200 Γ323 + Γ300 Γ030 + Γ300 Γ131 + Γ300 Γ232 + Γ300 Γ333 )
0 2
0 2
B0
(A )
2
(B 0 )2
A0
A0 B 0
(A )
∂ 2
=
+
+
+
+
+
+
−
∂r r
2B
4A2
r2
4B 2
4A2
Ar
4AB
B 00
B0 B0
A0
A0
=
−
+
−
2B
4B B
A
Ar
+
(2.10)
The set of surviving Ricci tensor terms are as follows:
R00
=
R11
=
R22
=
R33
=
B 00
B0 B0
A0
A0
−
+
−
2B
4B B
A
Ar
0
0
r
A
B
1
+
+ −1
2A A
B
A
r sin2 θ A0
B0
sin2 θ
+
+
− sin2 θ
2A
A
B
A
B 00
B 0 A0
B0
B0
−
+
+
−
2A 4A A
B
Ar
(2.11)
(2.12)
The solution to the Ricci tensor will yield the exact structure of the functions A(r) and B(r). After retrieving A(r)
and B(r) we have arrived at the Einstein Field Equations in a condensed form.
Rµν = −
8πG
c4
7
1
Tµν − gµν
2
(2.13)
To obtain the structure of functions A(r) and B(r), we will perform algebraic operations on our differential equations
to simplify the Ricci tensor structure.We divide R00 and R33 by A(r) and B(r) respectively and combine our results.
R00
R33
+
A(r) B(r)
=
+
=
0
B 00 (r)
B 0 (r)
B (r) A0 (r)
A0 (r)
B 00 (r)
−
+
− 2
−
2A(r)B(r) 4A(r)B(r) B(r)
A(r)
A (r)r 2A(r)B(r)
0
B 0 (r)
A (r) B 0 (r)
B 0 (r)
+
−
4A(r)B(r) A(r)
B(r)
A(r)B(r)r
0
−1
A (r) B 0 (r)
+
rA(r) A(r)
B(r)
(2.14)
Based on the fundamental assumptions of the Schwarzschild metric, we set the Ricci tensor solutions in the form of
partial differential equations equal to zero to satisy the vacuum condition.
0
A (r) B 0 (r)
−1
+
=0
rA(r) A(r)
B(r)
Z 0
Z
A (r)
−B 0 (r)
⇒
=
A(r)
B(r)
⇒
(2.15)
⇒ A(r)B(r) = C where C is an arbitrary constant of integration.
(2.16)
In order to acquire the proper relationship between A(r) and B(r) we must consider the geometry of space as our
spatial coordinates approach infinity. At infinity, the gravitational field will vanish thus creating the trivial flat
space-time condition. Thus,
lim A(r)
r→∞
=
1 and lim B(r) = 1
(2.17)
1
B(r)
(2.18)
r→∞
⇒ A(r) =
(2.19)
Since we have obtained the relationship between A(r) and B(r), we can substitute this relationship into one of the
remaining Ricci tensor equations to compute the exact structure of the functions. Allowing R11 to be in terms
of only B(r) yields, R11 = −1 + rB 0 + B = 0. Solving this differential equation yields B(r) = 1 +
is a constant of integration. Clearly, A(r) =
1
1+ r1
a
r
where a
. Because of the presence of the integration constant, we must
consider, as before, what happens to the spatial coordinates asymptotically. Analagous to the previous argument,
as r approaches infinity, flat-space time, also referred to as the Minkowski space-time, should be recovered. Due to
the presence of the extremely weak gravitational field, we consider Newton’s form of the time-only curvature that we
are analyzing. In a weak gravitational field, Newton describes the Minkowski metric as 1 +
2φ
c2
where φ =
GM
r
(the
gravitational potential in a weak field), c denotes the speed of light, G denotes the Newtonian gravitational constant,
8
and M denotes the mass of the gravitational body. We now utilize Newton’s definition to solve for the integration
constant a.
a
2φ
=1+ 2
r
c
a
2GM
⇒ = 2
r
c r
2GM
⇒a=
c2
B(r) = 1 +
(2.20)
(2.21)
We can now form the structure of the functions A(r) and B(r).
1
1 + 2GM
c2
2GM
B(r) = 1 +
c2
A(r) =
(2.22)
(2.23)
With the structure of the functions A(r) and B(r) known, we substitute the exact form back into the original line
metric and we have derived the Schwarzschild metric.
ds2 =
3
2GM
1+
c2
c2 dt2 −
1
1+
2GM
c2
!
dr2 − r2 dθ2 − r2 sin2 θdφ2
(2.24)
The Alcubierre Metric
After a thorough understanding of deriving the base case of the Schwarzschild metric we can extend our calculations
to additional line elements. In 1994, Miguel Alcubierre published a speculative paper on a possible solution to the
Einstein Field Equations. Alcubierre’s metric proposed a spacecraft that could travel at speeds exceeding that of
light. The speed of light would not be exceeded within its local frame of reference but rather would traverse distances
by contracting space in front of the craft and expanding space behind the craft. See Figure 3. It is important to
note that it is impossible for any given mass to accelerate faster than the speed of light. Thus, although the Eulerian
oberserver, one that remains at a fixed point in space, will perceive the full trajectory path as faster than that of
light, the individual on the spacecraft will always remain travelling inside his/her local light-cone. The Alcubierre
craft, in other words, would shift space allowing the spacecraft to arrive at its destination faster than light would in
normal space. This is done within the framework of general relativity and free of non-trivial wormholes. It can be
proven, in a similar fashion to that of the Schwarzschild metric, that the Alcubierre is consistent with the Einstein
9
Field Equations. Consider Miguel Alcubierre’s line metric:
ds2 = −(α2 − βi β i )dt2 + 2βi dxi dt + γij dxi dxj
(3.25)
In order to construct a metric that will utilize the contraction of space to push the ship, conditions must be met.
The lapse function, denoted by α measures the discrepancies between the proper time and coordinate time. We allow
this to equal 1. The shift vector β i , describes a mapping from one manifold to another preserving the topological
structure of the manifold. Without loss of generality, we can refer to this mapping as one from one hypersurface to
another as each hypersurface is a submanifold which thus inherits properties from the parent manifold. Therefore,
β x , representing this mapping of the x spatial coordinate, is equal to −vs f (rs ). The shift vectors β y and β z will be
set equal to 0. Finally, γij , which describes the geometry of the 3-metric hypersurfaces will be equal to the Kronecker
Delta function. The piecewise Kronecker Delta function is given by,
δij =


 0


i 6= j 

 1

i=j 
(3.26)
Given these substitutions, the original Alcubierre metric can be expressed as follows:
2
ds2 = −dt2 + (dx − vs f (rs )dt) + dy 2 + dz 2
(3.27)
where,
rs (t)
dxs (t)
dt
1
= (x − xs (t))2 + y 2 + z 2 2
f (rs )
=
vs (t)
=
(3.28)
(3.29)
tanh(σ(rs + R)) − tanh(σ(rs − R))
2 tanh(σR)
(3.30)
σ and R representing arbitrary parameters greater than zero.
Analagous to the calculations for the Schwarzschild metric, we can likewise express Miguel’s line metric in matrix
form.

gab




=



1
0
0
0
0
1
0
0
0
0
1
0
2vs f (rs )
0
0
vs2 f 2 (rs ) − 1
10









(3.31)
Solving for the inverse matrix yields,

g ab




=



1
0
0
0
0
1
0
0
0
0
1
0
−2vs f (rs )
vs2 f (rs )−1
0
0
1
vs2 f 2 (rs )−1









(3.32)
It can be shown the the surviving Christoffel symbols are as follows:
Γxtt
=
2vs f 0 (rs )rs0 t0 + 2vs0 f (rs ) − vs2 f (rs )f 0 (rs )rs0 t0
Γtxt
=
(vs2 f (rs )f 0 (rs )rs0 t0 − vs f 0 (rs )rs0 t0 − vs0 f (rs ))(
Γttt
=
(3.33)
−2vs f (rs )
1
+
vs2 f (rs ) − 1 vs2 f (rs )2 − 1))
−2vs f (rs )
1
(vs2 f (rs )f 0 (rs )rs0 t0 + vs vs0 f (rs )2 )( 2
+
)
vs f (rs ) − 1 vs2 f (rs )2 − 1
(3.34)
After solving for the surviving Ricci tensor equations, we obtain the following:
R00
=
((vs2 f (rs )f 0 (rs )rs0 − vs f 0 (rs )rs0 − vs0 f (rs ))(
(vs2 f (rs ) − 1)(−2vs f 0 (rs )rs0 ) + (2vs f (rs ))(vs2 f 0 (rs )rs0 )
(vs2 f (rs ) − 1)2
2vs2 f (rs )f 0 (rs )rs0
) + ((vs2 f 0 (rs )(rs0 )2 + vs2 f (rs )f 00 (rs )(rs0 )2 + vs2 f (rs )f 0 (rs )rs00 ) − (vs f 00 (rs )(rs0 )2 + vs f 0 (rs )rs0 )
(vs2 f 2 (rs ) − 1)2
−2vs f (rs )
1
− (vs0 f 0 (rs )rs0 ))( 2
+ 2 2
)
vs f (rs ) − 1 vs f (rs ) − 1
1
−2vs f (rs )
+ 2 2
))2
(3.35)
+ ((vs2 f (rs )f 0 (rs )rs0 − vs f 0 (rs )rs0 − vs f (rs ))( 2
vs f (rs ) − 1 vs f (rs ) − 1
−
R33
= −((2vs f 00 (rs )(rs0 )2 + 2vs f 0 (rs )rs00 + 2vs0 f 0 (rs )rs0 ) − (vs2 (f 0 (rs )rs0 )2 + vs2 f (rs )f 00 (rs )(rs0 )2 + vs2 f (rs )f 0 (rs )rs00 )
1
((v 2 f (rs ) − 1)(−2vs f 0 (rs )rs0 − 2vs0 f (rs ))(2vs f (rs ))(2vs vs0 f (rs )
(vs2 f (rs ) − 1)2 s
(2vs2 f (rs )f 0 (rs )rs0 + 2vs vs0 f 2 (rs ))
+ vs2 f 0 (rs )rs0 ) −
)) + ((vs2 (f 0 (rs )rs0 )2 + vs2 f (rs )f 00 (rs0 )2 + vs2 f (rs )f 0 (rs )rs0 r00
(vs2 f 2 (rs ) − 1)2
1
−2vs f (rs )
+
))
(3.36)
+ (vs0 )2 f 2 (rs ) + vs vs00 f 2 (rs ) + 2vs vs0 f (rs )f 0 (rs )rs0 )( 2
(vs f (rs ) − 1) vs2 f 2 (rs ) − 1
(v 2 f (rs ) − 1)(2vs f 0 (rs )rs0 ) − (2vs f (rs ))(vs2 f 0 (rs )rs0 ) 2vs2 f (rs )f 0 (rs )rs0
= ((vs2 f (rs )f 0 (rs )rs0 + vs vs0 f 2 (rs ))( s
−
))
(vs2 f (rs ) − 1)2
(vs2 f 2 (rs ) − 1)2
− ((vs2 f (rs )f 0 (rs )rs0 + vs vs0 f 2 (rs ))(
R31
+
(vs2 f 0 (rs )rs0 f 0 (rs )rs0 + vs2 f (rs )f 00 (rs )rs0 + vs2 f (rs )f 0 (rs )rs00
+
2vs vs0 f (rs )f 0 (rs )rs0 )(
1
−2vs f (rs )
+ 2 2
)
2
vs f (rs ) − 1 vs f (rs ) − 1
11
(3.37)
Due to the ambiguity of the spaceship’s trajectory function xs (t), we must ensure that no violations of the Einstein
Field Equations result. We must analyze the ship as a system existing within the constraints of general relativity.
Thus, we must ensure that this spacecraft remains in its respective light cone inside the manifold (the topological
geometry of space in n-dimensions) in this case one that is globally hyperbolic. The future and past light cones
represent the causal relationships between past, current, and future events. In other words, the location of events
in the past light cone will influence the position of the current event which in turn will influence the future event
positions as seen in figure (6). In terms of the spacecraft, the respective light cones define the boundaries of its
causal future and past positions. The light cone is comprised of all possible world lines or unique trajectories that
the ship can travel on. Recall that, although we do not know the structure of the trajectory function, we know
that the spacelike trajectory must be closed, implying the ship is traversing a distance. Closed trajectories can be
both timelike and spacelike, the first referring to seperate events differing in their location on the time axis while
the latter referring to event locations differing along one of the spatial axes. Clearly, when considering the path of a
ship we are observing a closed spacelike causal curve but not a closed timelike causal curve. The timelike trajectory
that the ship is travelling upon cannot be closed due to the globally hyperbolic structure. If the timelike trajectory
was closed, the ship would be in free fall. To ensure however that the world line the spacecraft travels on is in fact
a timelike curve, irrespective of the spaceship’s velocity (vs ), we let x = xs . This substitution, furthermore, allows
proper time to equal that of coordinate time. In the simplest of terms, coordinate time refers to time perceived by
an observer whereas proper time is defined by clocks at the position of the events. Because these two concepts of
time have now become equal, we can conclude no time dilation occurs. Thus the observer on the spacecraft does
not notice the speed approaching that of light. After this substitution is made, it becomes evident that the velocity,
vs = 1, and the acceleration, x00 = vs0 = 0. Without loss of generality, we moreover allow the arbitrary parameters σ
and R to equal 8 and 1 respectively. With this information, we can simplify the surviving Ricci tensor equations.
R00
=
((3(1 + e16 (4 + e16 (2 + e16 )2 )) − 2e3 2(−4cosh(16ρ)
+ cosh(32ρ)))2 (coth(8))2 (csch(8ρ))4 (tanh(8 − 8ρ) + tanh(8(1 + ρ)))2 )/
(
256e32 (1 + e16 + e32 + e16 cosh(16ρ))2 )
(3.38)
R33
=
0
(3.39)
R31
=
0
(3.40)
(3.41)
where ρ =
p
y 2 + z 2 . Notice that the expression for R00 , the only remaining equation due to our simplifying
assumptions, is algebraic in nature. Thus, the solution to Miguel Alcubierre’s, provides the exact geometry of space
12
surrounding the spacecraft despite the trajectory. We solve for ρ numerically to obtain this geometry. Doing so
yields an expression for ρ in terms of the y and z spatial coordinates.
y 2 + z 2 = ρ2 = 1.06984
(3.42)
It is evident, from the structure of the solution, that the spacecraft is travelling through space that resembles that
of a cylinder, whose circular cross sections have a radius of 1.03433 astronomical units.
Despite the mathematical consistency with the Einstein Field Equations, this solution is not physically meaningful.
The proposed line element violates energy laws that require,for all observers, the energy density to be positive. Inside
the manifold that the spacecraft is present, exist submanifolds commonly referred to as hypersurfaces. If we consider
measurements by observers whose 4-velocity (vector of velocity existing in 4-dimensional space-time) is normal to
the hypersurfaces that the ship travels upon, it can be shown that the energy density is given by,
E=
−1 vs2 ρ2 df 2
(
)
8π 4g 2 rs2 drs
(3.43)
where g is the determinant of the metric tensor matrix.
This expression implies that the energy density, as measured by the definition of observers from above, is always
negative. Thus the energy conditions are violated. Therefore, the only way to construct such a spacecraft would be
through the utilization of exotic matter. Since no evidence suggests the possibility of such matter existing we must
dismiss this faster than light solution to the Einstein Field Equations. Despite the failure of the metric to create
a warp drive that could physically exist, we are able to derive the York Time of the potential warp drive, which
measures the degree to which space expands and contracts. The York Time is defined to be
θ = vs
xs df (rs )
rs drs
(3.44)
Due to the simplifying assumptions made by Alcubierre, the York Time can be simplified to be
θ=
x df (rs )
ρ drs
(3.45)
Plotting this θ function against x and ρ yields the figures below. Figure 3 shows the results of Miguel Alcubierre.
Figures 4 and 5 depict our results in two different programming languages. The geometry differs slightly but the
expansion and contraction of space can still be observed.
13
4
Conclusion
In this paper we have considered the base case for the Einstein Field Equations, the Schwarzschild Metric. Through
the development of the Christoffel symbols, the Riemann tensor, and the Ricci tensor we have shown that the
gravitational field outside of a spherical, uncharged, non-rotating mass does indeed satisfy the Field Equations. We
furthermore introduce Miguel Alcubierre’s line metric, that attempts to describe the geometry of space surrounding
a spacecraft equipped with a warp drive. After a similar derivations of the respective Christoffel symbols, Riemann
tensor, and Ricci tensor we find that although the Field Equations are satisfied, energy laws are violated. This gives
rise to the need of exotic matter to make this metric physically significant. We will now begin to extend this research
to develop a new metric that satisfies the Einstein Field Equations but does not violate energy laws. This metric
will be reminiscent of both Alcubierre’s metric and the extension of Harold White’s work. We will attempt to find
the most efficient geometry of the warp drive that will provide a basis for faster than light travel. To do so, we will
utilize optimization techniques to minimize the amount of energy required and find the respective geometry that will
allow such a phenomenon.
References
[1] Markus
Hanke,
Solving
the
Einstein
Field
EquationsThe
Science
Forum,
(2012),
Wrinkles,
(1995)
http://www.thescienceforum.com/physics/30059-solving-einstein-field-equations.html.
[2] Larry
Smarr,
The
Einstein
Field
EquationsScience
and
Industry
http://archive.ncsa.illinois.edu/Cyberia/NumRel/EinsteinEquations.html.
[3] K. Schwarzschild,
ÃIJber das Gravitationsfeld eines Massenpunktes nach der Einstein’schen Theo-
rieSitzungsberichte der KÃűniglich Preussischen Akademie der Wissenschaften 1: (1916) 189âĂŞ196
[4] Susan
Larsen,
Lots
of
Calculations:
Relativity
Demystified
Tensor
Calculus
(2012),
http://physicssusan.mono.net/upl/9111/Lotsofcalculationsp.1326.pdf
[5] Leonard Parker, Curvature and the Einstein Equation http://scipp.ucsc.edu/ dine/ph171/171_8-3.pdf.
[6] multiple authors, Alcubierre Metric http://en.wikipedia.org/wiki/Alcubierre_drive
[7] multiple authors, Schwarzshild Metric http://en.wikipedia.org/wiki/Schwarzschild_metric
[8] multiple authors, The Einstein Field Equations http://en.wikipedia.org/wiki/Einstein_field_equations
[9] Miguel Alcubierre, The Warp Drive: Hyper-Fast Travel Within General Relativity Class. Quant. Grav. 11,
L73-L77 (1994)
14
Figure 1: Geometric interpretation of the four dimensional spherical coordinate system for the Schwarzschild metric
15
Figure 2: Geometric interpretation of the Schwarzschild metric.
Figure 3: Geometric interpretation of the Alcubierre metric.
16
17
5. ¥ 10-6
0.10
0
0.05
-5. ¥ 10-6
-0.10
0.00 r
-0.05
-0.05
0.00
x
0.05
0.10
Figure 5: Mathematica
18
-0.10
Figure 6: Visual Representation of a Light Cone.
19