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European Journal of Applied Sciences – Vol. 12, No. 3
Publication Date: June 25, 2024
DOI:10.14738/aivp.123.17054
Segman, Y. (2024). Modification in Alcubierre Warp Drive Geometry to Preserve Positive Energy Density. European Journal of
Applied Sciences, Vol - 12(3). 189-206.
Services for Science and Education – United Kingdom
Modification in Alcubierre Warp Drive Geometry to Preserve
Positive Energy Density
Yosef (Joseph) Segman
Independent Researcher Israel
ABSTRACT
Alcubierre introduced the warp drive model in 1994. However, Alcubierre warp
drive model resulted in negative energy density everywhere. The purpose of this
paper is to show that with a small modification in the Alcubierre warp drive
geometry, while keeping the warp drive bubble shape unchanged, we achieved a
total positive energy density.
Keywords: Energy Density, warp drive, Alcubierre warp drive bubble, Exotic Energy,
WEC, Matter, Complementary Matter (Dark Matter), Energy, Complementary energy.
INTRODUCTION
Alcubierre introduced the warp drive model in 1994 [1]. However, in line with the energy
conditions in general relativity, the Alcubierre model resulted in a negative energy density
everywhere, a problematic requirement since it involves exotic matter/energy.
When discussing the requirement for positive energy density, it is pertinent to draw from the
famed energy conditions of general relativity, particularly focusing on the weak energy
condition (WEC), which asserts that for any time-like vector, the energy momentum tensor's
contraction with this vector must be non-negative. Paradoxically, the Alcubierre warp drive
model, as originally proposed, violates the WEC by postulating a negative energy density in the
warp bubble region. The challenge, therefore, lies in modifying the warp drive's geometry so
that it obeys the known energy conditions, which is a nontrivial attempt to overcome the
theoretical energy density condition.
E. W. Lentz [2] suggested solitons to preserve the WEC, however, it is not yet clear that Lentz’
work preserves the WEC.
Y.J Segman [3], has explored mathematical modifications to the warp drive related to shifted
spatial coordinate metrics that allow for warp bubbles sustained by positive energy densities.
These alternative models are integral to enhancing our understanding, as they bring theoretical
speculation into the realm of testable science.
Furthermore, it is imperative to survey the fundamental principles that underpin the concept
of positive energy densities, such as the quantum inequalities proposed by Ford and Roman [4],
which place limits on the magnitude and duration of negative energy densities. Although such
inequalities may seem to open the door to cautiously optimistic speculation, we think that the
consequences of negative energy density on the spaceship fuselage could be devastating.
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European Journal of Applied Sciences (EJAS) Vol. 12, Issue 3, June-2024
Therefore, our aim is to look for ways to show that negative energy density everywhere is not
a must, and in fact this paper shows that positive energy density is achievable, although it is not
everywhere, yet the total energy density is positive.
Warp drive theory and the exotic energy including wormhole passages are discussed widely in
the literatures [5-25]. Yet, as noted above, the purpose of this paper is to show that one may
modify Alcubierre geometry without changing Alcubierre warp drive geometrical bubble while
preserving positive energy density.
Nonetheless, warp drive is not the only potential way to travel in space. Other feasible ways
presented by the author [26,27] have discussed spacetime jump drive. The idea of a jump drive
is based on the Fourier Transform. A stargate like device where the internal object is
transformed into the frequency domain by the Fourier transform while simultaneously the
restoration is constructed at the required spacetime location. Since the frequencies
representing the object are no longer part of any space-time universe, theoretically, one may
restore the object in any chosen space-time universe.
PRELIMINARIES AND ENERGY DENSITY
The 4D space-time metric is expressed by the line element, C.W. Misner [28]
ds2 = Tijdxidxj = −(K − K
iKi)dt2 − 2Kidxidt + Aijdxidxj
(2.1)
Where Aij is agreeable to be positive definite for all values of t for generating hyperbolic
spacetime geometry. The following Eq. 2.1, the line element of a warp drive space-time
geometry over the Cartesian manifold M = R4 (J. Natario [29]) gets the following form,
ds2 = (dx − Xdt)
2 + (dy − Ydt)
2 + (dz − Zdt)
2 − dt2
(2.2)
Where (X, Y, Z) are bounded smooth functions that define the new coordinate system, [29] and
represents the shift coordinate system to be defined by the arbitrary bounded functions
(u1, u2, u3) . Furthermore, if a spacetime does not violate the strong or the weak energy
conditions it must be flat, Theorem 1.7, J. Natario [29]. The purpose of this paper is to show
that Alcubierre warp bubble geometry under certain modification suggested hereunder many
not violate the weak energy condition everywhere while preserving positive total energy
density.
The volume element associated with the Eulerian observers is the trace.
θ = ∇X = TrK = Ki
i = ∂iX
i = ∂xX + ∂yY + ∂zZ (2.3)
and the energy density E yields,
E =
1
16π
(
(3)R + K
2 − Kj
iKi
j
) (2.4)
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Segman, Y. (2024). Modification in Alcubierre Warp Drive Geometry to Preserve Positive Energy Density. European Journal of Applied Sciences, Vol
- 12(3). 189-206.
URL: http://dx.doi.org/10.14738/aivp.123.17054
where the intrinsic (3)R curvature vanishes for flat metric. The geometric portion of the energy
density E yields
K
2 − Kj
iKi
j = 2∂xX∂yY + 2∂xX∂zZ + 2∂yY∂zZ −
1
2
(∂xY + ∂yX)
2
−
1
2
(∂xZ + ∂zX)
2 −
1
2
(∂yZ + ∂zY)
2
(2.5)
Note: the second term of Eq. 2.5 is always negative while the first term may produce a positive
quantity; therefore, to gain potential positive quantity, it is important to avoid nullification of
any of the axes of the new coordinate system (X, Y, Z).
ALCUBIERRE CHOICE OF FUNCTIONAL DISTANCE
Alcubierre geometry has the following form,
X = vf(r), Y = Z = 0 (3.1)
where the functional distance r used by the Alcubierre model [1] is the Euclidean radial square
root,
r
2 = (x − xs
)
2 + (y)
2 + (z)
2
(3.2)
where r is bounded in the interval [0, R] As a result of Eq. 3.1 i.e., the nullification of Y and Z and
of Eq. 3.2 The use of second-order functional distance, the outcome is negative energy density
everywhere, i.e.,
E =
1
16π
(K
2 − Kj
iKi
j
) =
1
16π
(θ
2 − Kj
iKi
j
) = −
v
2
(x
2+y
2
)
32πr
2
(
df
dr)
2
(3.3)
We shall show that there are several options that lead to Eq. 2.5 to be nonnegative with a
feasible geometric warp drive structure.
A potential way to overcome negative energy density in the Alcubierre model (Eq. 3.1) is to
replace the distance function r (Eq. 3.2) by first-order distance function of the form set in Eq.
3.4 (or Eq. 3.7, Eq. 3.8 or Eq. 3.9) and to avoid nullification of the new coordinates Y and Z so
that we may potentially gain a sufficient positive energy density.
The following formulation shows an alternative geometry that is not negative everywhere
provides a total positive energy density in the Alcubierre model.
Let,
Ψxyz = u1
(x − x0, y, z) = cos((x − x0 + y)) + cos((x − x0 − z)) (3.4)
Ψy = u2
(y) = sin(y) (3.5)
Ψz = u3
(z) = sin(z) (3.6)