A Possible Origin for Warp Drive Metrics
This Section focuses on a warp drive metric seen at Modern Relativity, which reduces the negative energy of both the Alcubierre and Hiscock Warp Drives, by considering time dilation in the ship's frame. That metric is then applied to the Schwarzchild solution of the Einstein Field Equation in order to describe the possible origins for warp drive metrics. The later metric is then further described by an article of Ford and Roman to further the meaning of negative energy.
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The "Zcphysicsms" [David White] Warp Drive
The Hiscock paper modifies the Alcubierre Warp Drive by reducing it two-dimensions and by it by a generic Lorentz transformation A(r) [1]. The "Zcphysicsms" metric performs a similar task with a function of "C(ct, r)," this metric then makes it easy to discuss the energy-density in the ships frame so we are left with the relation [2]:
ds2={C(rs)2[vsr g(rs)]2}dt-2 vsr g(rs)dtdt'-dx'2-dy'2-dz'2
with coordinates
g00={C(rs)2[vsr g(rs)]2}
g01=g10=-vsr g(rs)
g11=g22=g33=-1.
The Schwarzchild Solution
Taking this metric in comparison to Schwarzchild space one begins with
ds2=A'(r')dt'2-B'(r')dr'2r'2sin2q'df'2
thereby we rewrite the first equation with
g00={1-(2GM) C(rs)[vsr g(rs)]2}
g01=g10=-vsr g(rs)
g11=-{1-(2GM/r)}
g22=g33=-1
An arbitrary coordinate transformation of order r(b2(r))1/2-->r' is then applies, so that in cylindrical coordinates one has a solution of order
ds2={1-(2GM)'C'(r's)[vsr' g'(r's)]2}-{dr'2/1-(2GM/r)'}-r'2 2df'2-dz'2.
The Schwarzchild space is simply a solution to the Einstein Tensor Gab in spherical coordinates. From the definition of a Riemannain tensor
R00=(1/2)[T00+T11+T22+T33]
the origin of this warp drive metric comes from the energy-density and momentum of the stress-energy tensor. The problem is that the Schwarzchild metric describes a static space thus it can not describe angular momentum. Although the angular momentum can not be calculated through the Schwarzchild metric the energy-density still can be calculated which is given from the time dilation factor:
T00=-(b2/4)(c4/8pG)(dg/dr)(rq2/A(ct, r)).
which yields an energy-density of
T00=-[1/a2][vs2/4][sinq2][dg(rs)/drs]2.
It is likely that T11 T22 T33 terms for angular momentum can be solved through the Kerr metric, but we leave this open for the time being (since this would complicate the equations greatly). The negative energy of this space is similar to that generated by TabUaUb=>0, and may vanish for special case transformations.
The Generalized Second Law
A work on scalar fields by Ford and Roman [3] can explain where the negative energy of warp drive comes from. The Ford-Roman paper implies that the Generalized Second Law (GSL) is always satisfied in the case of a black hole:
Dtot=DS+DSmat=>0.
Where S is the entropy given by the area of the horizon. Now according to the Swarzchild Warp Drive Metric produced earlier matter may temporally become tachyoic within a black hole, and leave through the warp drive metric. This means mass from the black hole can interact with the Hawking radiation produced by the horizon (thus causing a gross violation of the conservation of energy) therefore nullifying the GSL:
Dtot=DS+DSmat=1, then the negative energy disappears from flat spacetime. The origin of the negative energy comes from an inertial frame, its appearance arises because the observer is a rest with the inertial frame. Therefore externally the negative energy exist near the Schwarzchild radius. Coincidentally the Ford-Roman paper describes the energy tensor in a Schwarzchild space given in radial coordinates by:
Ttr=(1/1-8pxf2)[(1-2x)f,t f,r*-2x(ff,tr -1/2(2M/r2)ff,t)]
A change in a black holes mass from scalar fields can thus be given by:
M*=(M2/2p) Int dW{[ln(1/1-8pxf2)],vv-1/4M[ln(1/1-8pxf2)],v}+ positive quantity
where v=t+r*. Thus the the momentum of a black hole can changes the total entropy of the black hole proportional to the initial mass Mi and current mass mc; Dtot=Mi-mc/t, thus the GSL is time dependent! The reason why the GSL holds in the Ford-Roman work because the negative energy of the scalar field amplifies the negative energy within the black hole thus compensating for Hawking radiation. Therefore the black hole not only absorbs a scalar field, but is actually capable of producing a scalar field as a by product of angular momentum, which causes entropy to increase. As the mass becomes tachyonic positive energy is absorbed by the horizon decreasing S, which intern causes the future horizon. That is to say as the negative energy flows into a black hole S increases, so the black hole ejects the excess energy in the form of tachyonic matter resulting in an Inverse Generalized Second Law (IGSL).
On scalar fields and entropy
The entropy of a static black hole under this scalar field is given through
S=16pM2/4(1-8px)-1
where x is the Einstein frame and f represents a Jordan frame. The argument is that if x=0 for the Einstein frame it must assume f=0 for the Jordan frame, this is made possible by a field redefinition which makes the Lagrangian xRf 2 term vanish. According to Ford and Roman the f term acts to preserve the GSL by acting on a black holes future horizon. Therefore the scalar field acts to increase S as a large negative flux enters, however the future horizon can save S for small fluxes.
Generalized Conclusions
Hawking radiation requires a black hole to have a negative entropy, the influx of scalar negative energy increases S, thus to restore the original entropy positive energy is absorbed. Therefore in a static black hole when a negative energy flux encounters the horizon its entropy increase temporally but the inflow of positive energy counters the effect preserving the GSL. With a rotating mass the angular momentum produces the negative energy by a violation of the WEC, in which the flux of positive energy is unable to conserve the GSL. Therefore the excess negative energy is ejected, resulting in the production of a warp drive metric (or wormhole) thereby satisfying the IGSL (which is an effect one would expect from Hawking radiation). When coupled to a scalar field this would be limited, the ejected matter of the warp drive would be limited to particles violating the WEC possessing two-dimensional radial angular momentum.
-The primary difference from the arguments in this page in relation to the Ford-Roman article is that the black hole itself is capable of producing a scalar field.
References
[1] Hiscock W. Quantum Effects in the Alcubierre warp drive spacetime. Class.Quant.Grav. 14 (1997) L183-88. gr-qc/9707024
[2] "Zcphysicsms" Modern Relativity
[3] Ford L and Roman T. Classical Scalar Fields and the Generalized Second Law. gr-qc/0009076
"Perhaps a Star Trek experience within our lifetime is not such a remote possibility." These are the words of Dr. Harold "Sonny" White, the Advanced Propulsion Theme Lead for the NASA Engineering Directorate. Dr. White and his colleagues don't just believe a real life warp drive is theoretically possible; they've already started the work to create one.
"There is hope," Harold "Sonny" White of NASA's Johnson Space Center said here Friday
The question can't be answered with current math, can it?
Originally posted by powerdrone
Question... Would this 'warp field' negate the effects of time dilation?
news.yahoo.com...
He found in that case, the warp drive could be powered by a mass about the size of a spacecraft like the Voyager 1 probe NASA launched in 1977.