M-Theory, Brane-World Cosmology, and Gauss-Codazzi Analysis

By George Shiber Sunday, December 6, 2015Saturday, August 17, 2019 In Cosmology, Grand Unification Physics, M Theory, Mathematical Physics, Philosophy Of Physics, Physics, Super String Theory, Supersymmetry Branes, calabi yau manifold

The imagination of nature is far, far greater than the imagination of man ~ Richard Feynman

In this post, I will derive universal expansionary acceleration in a braneworld context without any mention of exotic matter, hence further strengthening the explanatory power and success of M-theory. Just keep your eyes on

$\Im = \sum\limits_i {d{\theta _i}} \otimes d{\overline \theta ^i}\wp \left( {{{\left| \theta \right|}^2}} \right)$

and

${\not K_{aa,4}} - {\not K_{aa}}\frac{{\dot a}}{a} - a\dot a\left( {\delta _a^ \bot \delta _b^ \bot + {f^2}\delta _a^2\delta _b^2 + {f^2}{{\sin }^2}\theta \delta _a^3\delta _b^3} \right){\not K_{44}}$

throughout. The universe by all theoretical and phenomenological considerations, is undergoing an accelerated expansion (see: S. Perlmutter, et al., Nature, 391; also here and here) that indicates the presense of an energy component characterized by a negative pressure and the standard quantum cosmological explanations via the dynamics of dark energy have no justification beyond phenomenological ones, and so are unacceptable because they hence are not intrinsic parts of any law of nature: a metaplectic geometric equation that describes it as part of an algebraic solution. Why M-theory is the obvious question: not only can braneworld cosmology do that, but also in a way consistent with solving the hierarchy problem of fundamental interactions. The main reason is that braneworld supergravity differs from Einsteinian gravity in the following key way: gravitons can propagate in the higher-dimensional brane-bulk in a way that keeps the 3 other forces confined to the four-dimensional brane-manifold. There are many braneworld cosmologies I can work with, some in five dimensions defined in an anti-de Sitter bulk with a boundary terms in the action or the Randall–Sundrum one by regarding the brane as a single boundary of the bulk, with or without mirror symmetries, and a lot more such. What all have in common is dealing with the interaction between the bulk and the brane in a non-trivial way. To see this, note that when a gravitational wave or graviton crosses the brane-world it is subjected to hyper-deviation expressed in terms of the extrinsic curvature ${K_{ij}}$ of the embedded Kähler geometry group-theoretically representing the meta-tangent components of the local variation of the normal ‘unit vector’ (on that in a bit) and is expressed as

${K_{ij}} = \, - \frac{1}{2}\alpha _5^2\left( {T_{ij}^m - \frac{1}{3}{T^m}{g_{ij}}} \right)$

with ${\alpha _5}$ a constant proportional to the bulk gravitational constant and $T_{ij}^m$ the energy–momentum tensor of confined matter. The key now is to analyse the extrinsic curvature to the Friedmann–Robertson–Walker model since

${K_{ij}} = \, - \frac{1}{2}\alpha _5^2\left( {T_{ij}^m - \frac{1}{3}{T^m}{g_{ij}}} \right)$

is essentially an algebraic statement on the behavior of the extrinsic curvature in terms of the energy–momentum tensor of the confined sources. So we get FRW line

$d{s^2} = \, - d{t^2} + {a^2}\left[ {d{s^2} + f(r)\left( {d{\theta ^2} + {{\sin }^2}\theta d{\varphi ^2}} \right)} \right]$

with

${d{s^2} + f(r)\left( {d{\theta ^2} + {{\sin }^2}\theta d{\varphi ^2}} \right)}$

central for continuity and $f(r) = \sin r$ , $\sinh r$ corresponding to

$K = \left\{ {\begin{array}{*{20}{c}}1\\0\\{ - 1}\end{array}} \right.$

respectively. Hence, it follows that space–time can be embedded into a five-dimensional flat space inheriting the bulk-topology. Now I can flesh the bulk Riemann tensor

$^5{\not R_{\mu \nu \rho \sigma }} = \frac{{{\Lambda _5}}}{6}\left( {{\wp _{\mu \rho }}\,{\wp _{\nu \sigma }} - \,{\wp _{\mu \sigma }}{\wp _{\nu \rho }}} \right)$

with ${\wp _{\mu \nu }}$ the bulk metric and ${\Lambda _5}$ a bulk cosmological constant which can be positive, negative or zero in the flat bulk case. The brane-bulk embedding is determined by the components ${\Im ^\mu }$ of a map

$\Im :{V_4} \to {V_5}$

where

$\Im = \sum\limits_i {d{\theta _i}} \otimes d{\overline \theta ^i}\wp \left( {{{\left| \theta \right|}^2}} \right)$

and ${V_{...}}$ the metaplectic potential. So, the bulk vielbein $\left\{ {\Im _i^\mu ,{\eta ^\mu }} \right\}$ is

${\wp _{\mu \nu }} = \left( {\begin{array}{*{20}{c}}{{g_{ij}}}&0\\0&{{g_{55}}}\end{array}} \right)\quad \quad {g_{55}} = 1$

Now, the Gauss and Codazzi equations for the embedding in five dimensions are respectively

$\begin{array}{c}{\widetilde R_{ijkl}} = \left( {{K_{ik}}{K_{jl}} - {K_{il}}{K_{kl}}} \right) + \\^5{{\not R}_{\mu \nu \rho \sigma }}\Im _i^\mu \,\Im _j^\nu \,\Im _k^\rho \,\Im _l^\alpha \end{array}$

and

${K_{ij;k}} - {K_{ik;j}} = 0$

and the components of the extrinsic curvature are given by

${K_{ij}} = \, - \eta _i^\mu \Im _j^\nu {\wp _{\mu \nu }}$

Now, since the embedding is regular I can derive from

$\Im = \sum\limits_i {d{\theta _i}} \otimes d{\overline \theta ^i}\wp \left( {{{\left| \theta \right|}^2}} \right)$

the inverse Kähler expression

${g^{ij}}\Im _i^\mu \Im _j^\nu = {\wp ^{\mu \nu }} - {\eta ^\mu }{\eta ^\nu }$

hence getting the contractions of

$\begin{array}{c}{\widetilde R_{ijkl}} = \left( {{K_{ik}}{K_{jl}} - {K_{il}}{K_{kl}}} \right) + \\^5{{\not R}_{\mu \nu \rho \sigma }}\Im _i^\mu \,\Im _j^\nu \,\Im _k^\rho \,\Im _l^\alpha \end{array}$

with

${g^{ij}}$

$^5\not R = \widetilde R - \left( {{K^2} - {h^2}} \right) + {2^5}{\not R_{\mu \nu }}{\eta ^\mu }{\eta ^\nu }$

where $h = {g^{ij}}{K_{ij}}$ refers the mean curvature of the brane-world and ${\widetilde K^2} = {\not K^{ij}}{\not K_{ij}}$

Thus, the Einstein–Hilbert Lagrangian of the bulk decomposes as

$\begin{array}{l}^5\not R\sqrt { - \wp } = \widetilde R\sqrt { - g} - \left( {{K^2} - {h^2}} \right)\\ \cdot \sqrt { - g} + {2^5}{{\not R}_{\mu \nu }}{\eta ^\mu }{\eta ^\nu }\sqrt { - g} \end{array}$

The Euler–Lagrange equations with respect to ${g_{ij}}$ gives the brane equations of motion

$\begin{array}{l}{{\not R}_{ij}} - \frac{1}{2}\not R{g_{ij}} = {{\not Q}_{ij}} + \\\left( {^5{{\not R}_{\mu \nu }} - {{\frac{1}{2}}^5}\not R{\wp _{\mu \nu }}} \right)\\ \cdot \,\Im _i^\mu \,\Im _j^\nu + 8\pi \,G\,T_{ij}^m\end{array}$

with

${\not Q_{ij}} = {g^{mn}}{\not K_{im}}{\not K_{jn}} - h{\not K_{ij}} - \frac{1}{2}\left( {{{\not K}^2} - {h^2}} \right){g_{ij}}$

Note, this quantity is defined by the extrinsic curvature and it does not exist in Einstein’s equations as defined in pure Riemannian geometry.

Now we can derive

$\left( {^5{{\not R}_{\mu \nu }} - {{\frac{1}{2}}^5}\not R{\wp _{\mu \nu }}} \right)\Im _i^\mu \Im _j^\nu = - {\Lambda _5}{g_{ij}}$

which implies a deep conclusion

$\widetilde {\not \not \not Q}_\mu ^{\mu \nu } = 0$

Now, in order to give a pure brane-geometric description of dark energy, realize that in five dimensions Codazzi’s equation can be solved separately, so, denoting the spatial indices in the brane by the letters a, b, c, d = 1,…,3, we get the following separability

${\not K_{aa,c}} - {\not K_{ad}}\Gamma _{ac}^d = {\not K_{ac,a}} = - {\not K_{cd}}\Gamma _{aa}^d$

and

The first equation gives ${\not K_{11,c}} = 0$ so it follows that ${\not K_{11}}$ is a function of $t$ only, $b(t)$ , which is the radial bending of the braneworld. Hence, we get

${\not K_{44}} = - \frac{1}{{\dot a}}\frac{d}{{dt}}\left( {\frac{b}{a}} \right)$

Applying the same arguments for ${\not K_{44}}$ and ${\not K_{33}}$ , we can derive the general solution

$\left\{ {\begin{array}{*{20}{c}}{{{\not K}_{ab}} = \frac{b}{{{a^2}}}{g_{ab}}}\\{{{\not K}_{44}} = - \frac{1}{{\dot a}}\frac{d}{{dt}}\left( {\frac{b}{a}} \right)}\end{array}} \right.$

Letting $B = \dot b/b$ and the Hubble parameter $H = \dot a/a$ , we get

$\left\{ {\begin{array}{*{20}{c}}{{{\not Q}_{ab}} = \frac{{{b^2}}}{{{{\dot a}^4}}}\left( {2\frac{B}{H} - 1} \right){g_{ab}}}\\{{{\not Q}_{44}} = - \frac{{3{b^2}}}{{{{\dot a}^4}}}}\end{array}} \right.$

After Fredholm substitution and Gaussian elimination, we obtain the Friedmann equation as modified by the presence of the extrinsic curvature and the bulk constant curvature ${\Lambda _5}$

${\dot a^2} + k = \frac{{8\pi G}}{3}\rho {a^2} + \frac{{{\Lambda _5}}}{3}{a^2} + \frac{{{b^2}}}{{{a^2}}}$

with the radial bending ${\not K_{11}} = b(t)$ remains arbitrary. In the context of the use of junction conditions, the above expression can be tested for compatibility with the use of the Israel–Lanczos condition

${K_{ij}} = \, - \frac{1}{2}\alpha _5^2\left( {T_{ij}^m - \frac{1}{3}{T^m}{g_{ij}}} \right)$

applied to the solution

After calculating ${\not K_{11}} = b$ , one finds that

$b(t) = - \frac{1}{6}\alpha _5^2\rho {a^2}$

Replacing this in

${\dot a^2} + k = \frac{{8\pi G}}{3}\rho {a^2} + \frac{{{\Lambda _5}}}{3}{a^2} + \frac{{{b^2}}}{{{a^2}}}$

one gets the Friedmann equation with the square of the energy density

$\begin{array}{l}{{\dot a}^2} + k = \frac{{8\pi G}}{3}\rho {a^2} + \frac{{{\Lambda _5}}}{3}{a^2} + \\\frac{{\alpha _5^4}}{{36}}{\rho ^2}{a^2}\end{array}$

thus clearly contradicting the standard view that Friedmann’s equation in braneworlds does not necessarily imply the presence of a ${\rho ^2}$ term, and this fact follows given the Israel–Lanczos junction condition. To proceed with this geometrical interpretation let us view ${\not K_{11}}$ as a cosmic ‘fluid’ with energy–momentum tensor

${\tau _{ij}} \equiv - \frac{{{{\not Q}_{ij}}}}{{8\pi G}}$

As a consequence of $\not Q_\nu ^{\mu \nu } = 0$ , it follows that this cosmic ‘fluid’ does not exchange energy with the ordinary confined matter. Letting ${p_b}$ be its pressure and ${\rho _b}$ the corresponding energy-density, one can then represent ${T_{ij}}$ as

${\tau _{ij}} = \left( {{p_b} + {\rho _b}} \right){U_i}{U_j} + {p_b}{g_{ij}}$

with ${U_i} = \delta _i^4$ to which one can add a state-like equation

${p_b} = \left( {{\gamma _b}(t) - 1} \right){\rho _b}$

with ${\gamma _b}(t)$ a Heisenberg functional of time. Now, we can derive

$\begin{array}{l}\not Q = {g^{ij}}{{\not Q}_{ij}} = 3{p_b} - {\rho _b} = \\{g^{ij}}{{\not Q}_{ij}} = \frac{{6{b^2}}}{{{a^4}}}\frac{B}{H}\end{array}$

Replacing ${p_b}$ in the trace expression and using the equation of state we obtain the equation for $b(t)$

$\frac{{\dot b}}{b} = \frac{1}{2}\left( {4 - 3{\gamma _b}(t)} \right)\frac{{\dot a}}{{a\dagger }}$

which is quasi-morphic to the x-matter: the main phenomenological candidate for dark energy, implying that the brane extrinsic curvature should be the fundamental explanation for such models. Taking ${\gamma _b}$ as constant, we get the solution

$b(t) = {b_0}a{(t)^{\frac{1}{2}\left( {4 - 3{\gamma _b}} \right)}}$

with ${b_0}$ the metaplectic integration constant. So, the ‘bending’ energy becomes

${\rho _b} = \frac{{3{b_0}}}{{8\pi G}}{a^{ - 3{\gamma _b}}}$

Consider now the simplest estimate, a vanishing ${\Lambda _5}$ corresponding to a flat bulk and a spatially flat FRW-braneworld composed mainly of dark matter and the bending contribution in place of the dark energy. Letting

${\Omega _m} \equiv \frac{{8\pi G{\rho _m}}}{{3{H^2}}}$

for dark matter,

${\Omega _b} \equiv \frac{{8\pi G{\rho _b}}}{{3{H^2}}}$

for the geometric contribution, the partial deceleration parameter is

$q = - \frac{{\ddot a\dot a}}{{{{\dot a}^2}}} = \left( {3{\gamma _b} - 2} \right)\frac{{{\Omega _b}}}{2} + \frac{{{\Omega _m}}}{2}$

Hence, for ${\Omega _m} \sim 0.3$ and ${\Omega _b} \sim 0.7$ , a ‘present’ cosmological time entails a universe driven by the extrinsic curvature occurs whenever ${\gamma _b} < 0.52$ as in the x-matter case. The geometrical interpretation for the x-matter allows us to consider observational tests as measurements of the extrinsic curvature and its evolution for constant values of ${\gamma _b}$ . This has been an inversion with respect to the usual approaches in the crucial sense that the data may be used to measure the evolution of geometry, and such results greatly facilitate the elaboration of theories explaining the bending of the braneworld, or, in other words, the evolution of the dark energy component. Again: M-theory is ‘M’-agic!