[Paper Review] Embedding General Relativity with varying cosmological constant term in five-dimensional Brans-Dicke theory of gravity in vacuum

Luz M. Reyes, José Edgar Madriz Aguilar|ArXiv.org|Feb 27, 2009

Cosmology and Gravitation Theories4 references3 citations

TL;DR

This paper proposes a five-dimensional Brans-Dicke theory of gravity in vacuum, showing that a four-dimensional general relativity with a time-varying cosmological constant can be geometrically induced on a 4D hypersurface when the 5D scalar field depends only on the extra coordinate. The key result is that the 4D gravitational coupling and cosmological constant emerge from the 5D geometry, with the induced cosmological constant matching observational values when the fifth dimension is on the order of the Hubble length.

ABSTRACT

We investigate the possibility to recover a four-dimensional (4D) general theory of relativity, as embedded in a 5D spacetime where gravity is governed by a five-dimensional (5D) Brans-Dicke (BD) theory of gravity. Employing the Gauss-Codazzi-Ricci equations and some ideas of the induced matter theory, we obtain that when the 5D BD scalar field is only depending of the extra coordinate, it is possible to recover on every generic 4D hypersurface the usual Einstein field equations plus a cosmological constant term, with matter sources described by the energy-momentum tensor of induced matter and where the cosmological constant under certain conditions can vary with time. Finally to illustrate the formalism, we give an example in which we induced a cosmological constant from warped product spaces.

Motivation & Objective

• To explore whether four-dimensional general relativity with a cosmological constant can be geometrically induced from a five-dimensional Brans-Dicke theory of gravity in vacuum.
• To investigate how the 4D gravitational coupling and cosmological constant emerge from the higher-dimensional geometry, particularly when the Brans-Dicke scalar field depends only on the extra coordinate.
• To demonstrate that the induced matter and cosmological term can reproduce the effective equation of state of a cosmological constant, even when the 4D cosmological constant varies with time.
• To provide a concrete example using warped product geometries where the induced 4D cosmological constant matches observational estimates.

Proposed method

• The Gauss-Codazzi-Ricci equations are applied to a 5D spacetime with a Brans-Dicke scalar field depending only on the extra coordinate to derive the induced 4D field equations.
• The 5D action is reduced to a 4D effective action, showing that the 4D Newton constant is related to the 5D Brans-Dicke scalar field via $16\pi G_N = 1/\varphi(\psi_0)$.
• The induced energy-momentum tensor for matter is derived from the 5D Ricci curvature and the scalar field profile, following the induced matter framework.
• A warped product metric is used as a concrete example to compute the induced 4D cosmological constant, with $g_{\psi\psi}$ as a function of the extra coordinate.
• The effective equation of state parameter $\omega_{\text{eff}}$ is computed, showing $\omega_{\text{eff}} = -1$ when the induced matter has $\omega_{\text{IM}} = -1$, consistent with a cosmological constant.
• The value of the 5D scalar field $\psi_0$ is constrained to $6.12 \cdot 10^{26}$ m to match the observed $\Lambda_0 \simeq 10^{-54} \, \text{m}^{-2}$, linking it to the Hubble length.

Experimental results

Research questions

• RQ1 Can a four-dimensional general relativity with a time-varying cosmological constant be geometrically embedded in a five-dimensional Brans-Dicke theory of gravity in vacuum?
• RQ2 How does the 5D Brans-Dicke scalar field, dependent only on the extra coordinate, lead to a 4D effective theory with a cosmological constant and induced matter?
• RQ3 What is the relation between the 5D gravitational coupling and the 4D Newton constant in this embedding framework?
• RQ4 Can the induced 4D cosmological constant match observational values, and what does this imply for the size of the extra dimension?
• RQ5 What is the effective equation of state of the induced matter and cosmological term in the resulting 4D theory?

Key findings

• The 4D effective theory on a hypersurface $\Sigma_0: \psi = \psi_0$ reproduces the Einstein field equations with a cosmological constant term, where both the matter and the cosmological constant are geometrically induced from the 5D spacetime.
• The 4D Newton constant is induced via $16\pi G_N = 1/\varphi(\psi_0)$, linking the 4D gravitational coupling to the 5D Brans-Dicke scalar field evaluated at $\psi_0$.
• When the 5D metric component $g_{\psi\psi}$ is constant, the induced cosmological constant $\Lambda_0$ is time-independent; when $g_{\psi\psi}$ is time-dependent, $\Lambda(t)$ varies with time.
• For the warped product geometry example, the induced 4D cosmological constant is $\Lambda_0 = \frac{3}{8\psi_0^2}$, and setting $\Lambda_0 \simeq 10^{-54} \, \text{m}^{-2}$ yields $\psi_0 = 6.12 \cdot 10^{26} \, \text{m}$, comparable to the Hubble length.
• The effective equation of state parameter is $\omega_{\text{eff}} = -1$, indicating that the combined effect of induced matter ($\omega_{\text{IM}} = -1$) and the cosmological term mimics a pure cosmological constant.
• The value $\psi_0 \simeq 6.12 \cdot 10^{26} \, \text{m}$ is approximately 0.2 times the present Hubble length, suggesting a natural explanation for the fifth dimension's indirect observability.