Historical research guide · 0558 · 2021

Perfect fluid warp drive solutions with the cosmological constant

Osvaldo L. Santos-Pereira · Everton M. C. Abreu · Marcelo B. Ribeiro

In one minute

Miguel Alcubierre wrote his warp metric down in 1994 without ever solving Einstein’s equations for it — he chose a geometry and read off the stress-energy it would need, and the answer looked like negative energy. Osvaldo Santos-Pereira, Everton Abreu and Marcelo Ribeiro, working at Brazil’s federal universities in Rio de Janeiro, have spent a series of papers doing the missing work: take the same warp metric, put real matter on the other side of the equation, and solve. Here they add the cosmological constant — the vacuum energy term itself — with a perfect fluid as the source. Four families of solutions come out. Two reproduce their earlier vacuum result, in which the warp drive turns out to obey the Burgers shock-wave equation. Two are new, and in those the cosmological constant is exactly what turns the shift vector from a complex-valued function into a real one. Their conclusion is the headline: negative matter density may not be a strict requirement for warp speeds, but a complex, off-diagonal distribution of energy and momentum is.

Connection

Chapter 4 is metric engineering — you travel by changing the distance, not by pushing on propellant — and this paper is one of the few that actually solves Einstein’s equations for the Alcubierre geometry instead of guessing the source. It also joins chapter 2 to chapter 4 directly: the cosmological constant is the vacuum energy term, and here it is the term that makes the warp solution real-valued.

From the source

Once the cosmological constant is carried into the Einstein tensor, the energy density an Eulerian observer measures in the warp bubble is the cosmological constant minus a term built from the gradient of the bubble shape function — so the weak and dominant energy conditions can be satisfied if the cosmological constant is positive and large enough, where in Alcubierre’s original calculation without it they were violated.Section 2.3, Eqs. 2.24 and 2.25
Solving the full Einstein equations for the Alcubierre metric with a perfect fluid source and the cosmological constant gives four sets of partial differential equations: two, called 1b and 2b, force the cosmological constant, the density and the pressure all to zero and return the vacuum solution connected to the Burgers shock-wave equation; two, called 1a and 2a, are new and let the shift vector depend on time and one transverse coordinate.Section 3.2, Cases 1a, 1b, 2a and 2b; Table II
The two new solutions carry an equation of state tying the three quantities together — the cosmological constant equals two pi times the quantity three times the matter density minus the pressure.Eqs. 3.10 and 3.15; Section 6, first paragraph
The cosmological constant is what makes those solutions physically readable: with it the shift vector is a real-valued function provided the cosmological constant is at least eight pi times the matter density and at least eight pi times the pressure, and without it the same shift vector would be complex-valued.Eqs. 3.11, 3.12, 3.16 and 3.17; Section 6, second and third paragraphs
For the perfect fluid source the four energy conditions reduce to simple statements about the density and the pressure: as derived in the text, the weak and the dominant conditions hold when the matter density is positive, the strong condition when three times the pressure plus the density is not negative, and the null condition when the density plus the pressure is not negative. Table I lists the same four with the strong and dominant entries in the reverse order, so read the derivations in Sections 4.2 and 4.3 as the authoritative statement.Sections 4.1 to 4.4, Eqs. 4.2, 4.5, 4.6, 4.8 and 4.13; compare Table I
Taking this paper with the three before it, the authors state that negative matter density may not be a strict requirement for reaching global superluminal speeds, and conjecture that the real key to a superluminal propulsion system is a complex distribution of energy, matter and momentum — the shift vector along the direction of motion forces off-diagonal terms in the source, so dust alone will not do it.Section 6, final two paragraphs; Abstract, last sentence

The way in

https://doi.org/10.1140/epjp/s13360-021-01899-7

Historical source-access record. This guide links to the original; it does not reproduce the paper text.

TEXT. Full text below, abstract to references, from the CC BY arXiv version. The paper is set in LaTeX and the extraction flattened tensor indices, fractions and stacked derivatives, so the display equations are reset here in plain notation: indices are written in parentheses after the symbol, so G(00) and T(alpha beta); partial derivatives keep the ∂ symbol; the shift-vector abbreviation β equals minus β(1) equals v_s times f is used from Section 2.2 onward exactly as the authors define it. Inequalities are written in words because the page is MDX. The two summary tables are reset as lists. Reference numbers in the running text are kept in square brackets and resolve against the reference list at the foot.

How to cite it

Osvaldo L. Santos-Pereira, Everton M. C. Abreu, Marcelo B. Ribeiro (2021) Perfect fluid warp drive solutions with the cosmological constant doi:10.1140/epjp/s13360-021-01899-7

Where it sits in the curriculum

The metric, warp drives and wormholes