The Wheeler-DeWitt Metric, Indefiniteness, Quantum Cosmology, and Gravity

By George Shiber Wednesday, December 16, 2015Saturday, August 17, 2019 In Cosmology, Grand Unification Physics, M Theory, Mathematical Physics, Philosophy Of Physics, Super String Theory, Supersymmetry Jordan-Brans-Dicke field, Quantum Cosmology

Go down deep enough into anything and you will find mathematics ~ Dean Schlicter

The unification problem in physics has at its heart the problem of time: quantum physics, via the Heisenberg Uncertainty Principle between time and energy is not compatible with the Einstein notion of time as coupled with space that gives rise to a ‘smooth’ spacetime continuum. John Wheeler and Bryce DeWitt did successfully resolve this incompatibility via the Wheeler-DeWitt equation: however, at two high prices. One, time ceases to play any dynamical role in the universe since it occurs idly in the equation, and secondly, by metaplectic geometry, there can be no quantum cosmological universal metric definable: see below on that deep problem for defining a notion of time-directionality in quantum cosmology. In this post, I will try and analyse the ‘metric problem’ and leave my analysis of the Page-Wootters ‘solution’ to the WdW problem of time for another post. The metric-problem in quantum cosmological settings with quantum gravity is that it arbitrarily varies on the configuration space of canonical gravity. Let me dive in. The central role in canonical gravity is played by the Hamiltonian constraint, with $c = 1$ ,

$H = \int {{d^3}} xN\sqrt h \left( {\frac{{16\pi G}}{{\sqrt h }}{G_{abcd}}{\pi ^{ab}}{\pi ^{cd}} - \frac{{R - 2\Lambda }}{{16\pi G}} + {{\not H}^\dagger }} \right) = o$

with

$\frac{{16\pi G}}{{\sqrt h }}{G_{abcd}}{\pi ^{ab}}{\pi ^{cd}} - \frac{{R - 2\Lambda }}{{16\pi G}} + {\not H^\dagger }$

key as we shall see, and $R$ the Ricci scalar on a three-dimensional space, and $\Lambda$ the cosmological constant, and $N$ the lapse function.

The coefficients ${G_{abcd}}$ depend on the three-metric ${h_{ab}}$ and play the role of a metric on $Riem\Sigma$ , the space of all three-metric associated with a manifold $\Sigma$ . These are the DeWitt metric and are given by

${G_{abcd}} = \frac{1}{{2\sqrt h }}\left( {{h_{ac}}{h_{bd}} + {h_{ad}}{h_{bc}} - {h_{ab}}{h_{cd}}} \right)$

due to the Riemann-Ricci condition on $\Sigma$ , one modifies it via a parameter $\alpha$

$G_{abcd}^\alpha = \frac{1}{{2\sqrt h }}\left( {2\alpha {h_{ac}}{h_{bd}} + 2\alpha {h_{ad}}{h_{cd}}} \right)$

This gives the class of Wheeler-DeWitt metrics given the quantum version of the constraint

$H = \int {{d^3}} xN\sqrt h \left( {\frac{{16\pi G}}{{\sqrt h }}{G_{abcd}}{\pi ^{ab}}{\pi ^{cd}} - \frac{{R - 2\Lambda }}{{16\pi G}} + {{\not H}^\dagger }} \right) = o$

we have thus the WheelerDeWitt equation

$\widehat H\,\Psi = 0$

Notice, the Wheeler-DeWitt metrics above exhaust the class of all ultralocal metrics on

$\Sigma$

Hence, they do not contain space derivatives. The inverse metric is then

$G_\beta ^{abcd} = \frac{{\sqrt h }}{{2{\alpha ^\dagger }}}\left( {2\alpha {h^{ac}}{h^{bd}} + 2\beta {h^{ad}}{h^{bc}} - 2\beta {h^{ab}}{h^{cd}}} \right)$

with $\alpha + \beta = 3\alpha \beta$ . In the case of general relativity, which corresponds to the choice $\beta = 1\left( {\alpha = 1/2} \right)$ , the signature of

$G_{abcd}^\alpha = \frac{1}{{2\sqrt h }}\left( {2\alpha {h_{ac}}{h_{bd}} + 2\alpha {h_{ad}}{h_{cd}}} \right)$

is, at each point of space, given by ${\rm{diag}}\left( { - , + , + , + , + , + } \right)$ , hence vectors in $Riem\Sigma$ can be lightlike, that is, they may have vanishing scalar product with respect to

$G_\beta ^{abcd} = \frac{{\sqrt h }}{{2{\alpha ^\dagger }}}\left( {2\alpha {h^{ac}}{h^{bd}} + 2\beta {h^{ad}}{h^{bc}} - 2\beta {h^{ab}}{h^{cd}}} \right)$

On the other hand, there are sets in superspace where the infinitely many minus signs in DeWitt’s metric reduce to one global minus sign. The Wheeler-DeWitt equation is therefore truly hyperbolic, which has deep consequences for the arrow of time problem in quantum cosmology.

Now, the obvious question is, what happens for other values of $\beta$ ? Clearly there is a critical value ${\beta _c} = 1/3$ for which the betta metric is degenerate. In the case of $\beta < {\beta _c}$ we have positive definite and for $\beta > {\beta _c}$ indefinite. This is clear when one introduces new coordinates on $Riem\Sigma$

$\left\{ {\begin{array}{*{20}{c}}{\tau = 4\sqrt {\left| {{\beta ^\dagger }\frac{1}{3}} \right|} {h^{1/4}}}\\{{{\widehat \tau }_{ab}} = {h^{ - 1/3}}{h_{ab}}}\end{array}} \right.$

given that the line element in superspace can be written as

$\begin{array}{l}G_\beta ^{abcd}d{h_{ab}} \otimes d{h_{cd}} = \, - {\rm{sgn}}\left( {\beta - \frac{1}{3}} \right)d\tau \otimes d\tau \\ + \frac{{{\tau ^2}}}{{G\left| {\beta - \frac{1}{3}} \right|}}{\rm{Trace}}\left( {{{\widehat \tau }^{ - 1}}d\widehat \tau \otimes {{\widehat \tau }^{ - 1}}d\widehat \tau } \right)\end{array}$

We are now in a position to study the connection of the sign in the WheelerDeWitt metric with the attractivity of gravity as well as possible cosmological consequences. So let me say something about the sign of the acceleration of the whole three-volume

$V \equiv \int {\sqrt h } {d^3}x$

and being a coordinate-independent quantity, I thus need the second time derivative of $\sqrt h$ derived from the Hamilton equations of motion

${\dot h_{ab}} = \left\{ {{h_{ab}},H} \right\} = 32\varpi G{G_{abcd}}{\tau ^{cd}}$

and

$\begin{array}{l}{{\dot \varpi }^{ab}}\left\{ {{\varpi ^{ab}},H} \right\} = 16\varpi G\frac{{{{\not \partial }_{mnpq}}}}{{\not \partial {h_{ab}}}}{\varpi ^{nm}}\\ - \,\sqrt h {M^{ab}} - \frac{{\sqrt h }}{{16\pi }}\left( {{G^{ab}}\tau \Lambda {h^{ab}}} \right)\end{array}$

with

$\begin{array}{c}\sqrt h {G^{ab}} = \, - \frac{\delta }{{\delta {h_{ab}}}}\int {\sqrt h } R{d^3}x = \\\sqrt h \left( {{R^{ab}} - \frac{1}{2}{h^{ab}}R} \right)\end{array}$

and

$\sqrt h {M^{ab}} = \frac{\delta }{{\delta {{\dot h}_{ab}}}}\int {\sqrt h } {\not H^\dagger }{d^3}x = \frac{{\not \partial \sqrt h {{\not H}^\dagger }}}{{\not \partial {h_{ab}}}}$

$\frac{{\not \partial \sqrt h {{\not H}^\dagger }}}{{\not \partial {h_{ab}}}}$

being key to the matter Hamiltonian depending ultralocally on the metric.

Putting all of the above together, we can derive

$\begin{array}{l}\frac{{{d^2}V}}{{d{t^2}}} = \left( {3\alpha - 1} \right)\int {{d^3}} x \cdot \\\left( { - \frac{3}{8}{G^{abcd}}{{\dot h}_{ab}}{{\dot h}_{cd}} + \sqrt h {h_{ab}}\left( {{G^{ab}} + \Lambda {h^{ab}} + 16\widehat \tau G{M^{ab}}} \right)} \right)\end{array}$

Now, by using the constraint equation

${G^{abcd}}{\dot h_{ab}}{\dot h_{cd}} = 4\sqrt h \left( {R - 2\Lambda - 16\widehat \tau G{{\not H}^\dagger }_m} \right)$

We can therefore deduce that gravity is attractive if the sign in front of the integral is negative via the following

$\begin{array}{l}\frac{{{d^2}V}}{{d{t^2}}} = \, - 3\left( {3\alpha - 1} \right)\int {{d^3}} x\sqrt h \cdot \\\left( {\frac{2}{3}R - 2\Lambda 16\widehat \tau G\left( {{{\not H}^\dagger } - \frac{1}{3}{h^{ab}}\frac{{\not \partial {{\not H}^\dagger }}}{{\not \partial {h^{ab}}}}} \right)} \right)\end{array}$

This can be confirmed by analysis of terms in the above formula: gravity is attractive if a positive Ricci scalar contributes with a negative sign to the acceleration. This occurs when $\alpha > {\alpha _c} \equiv 1/3$ , which is the critical value for the metric

$G_{abcd}^\alpha = \frac{1}{{2\sqrt h }}\left( {2\alpha {h_{ac}}{h_{bd}} + 2\alpha {h_{ad}}{h_{cd}}} \right)$

given

${{{\not H}^\dagger } - \frac{1}{3}{h^{ab}}\frac{{\not \partial {{\not H}^\dagger }}}{{\not \partial {h^{ab}}}}}$

So a cosmological constant $\Lambda > 0$ acts repulsively. Thus, the attractivity of gravity is intimately connected with the indefinite signature of the Wheeler-DeWitt metric as can be attested by

${\not H^\dagger } = \frac{{\widehat \pi _\phi ^2}}{{2h}} + \frac{1}{2}2\alpha {h^{ab}}{\phi _{,a}}{\phi _{,b}}\,\frac{1}{2}{m^2}{\phi ^2} + V\left( \phi \right)$

with $\phi$ matter terms. The deep point now is that the Wheeler-DeWitt metrics have an indefiniteness that cannot allow us to integrally-measure any directionality of time as can be seen by the Friedmann equation for the scale factor ${a^\dagger }$

$\frac{{{{\dot a}^{{\dagger ^2}}}}}{{2\left( {3\alpha - 1} \right)}} = \, - \kappa + \frac{{8\pi G}}{3}{\dot a^{{\dagger ^2}}}\left( {\rho + \frac{\Lambda }{{8\pi G}}} \right)$

This is directly the problem of the attractivity of gravity as it intimately connects with the indefinite signature of the Wheeler-DeWitt metric, but indirectly points to a metric-measure problem for any cosmological directionality of time: for another post.

2 Responses

Arne
Sunday, December 20, 2015

Great work
- George Shiber
  Monday, December 21, 2015
  
  Too kind of you Arne, I followed Edward Witten’s advice when he told me there are no superstring/M-theorists bloggers on social media, I communicate with M-theorists via e-mail on a daily basis. Again, real pleasure to thank you, best wishes, George!