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Quantum densities in Curved Spacetime

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Adegbite Inioluwa Precious^*

This version is not peer-reviewed

Submitted:

01 July 2024

Posted:

02 July 2024

Withdrawn:

23 October 2024

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Abstract

Defining integrals over volume element in curved spacetime together with the mass and energy densities within that region of spacetime is apparently equivalent to the given energy and mass of that body or particle. With this in mind equations involving energies and masses can be rewritten in terms of their densities and the integral over the curved space-time volume element. The idea find its place in quantum mechanical frameworks systematically accounting for the effect of spacetime volume structures and gravity on dynamics of quantum systems. This article briefly explores these concepts and runs through various frameworks and equations relevant to this idea including the quantum stress energy tensor, and expressing Schrodinger’s equation in ways that are tied to general relativity it also discusses the cosmological constant and the cosmological constant problem.

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Subject: Physical Sciences - Theoretical Physics

Introduction

The energy density [1,2] refers to the amount of energy stored in a given volume of space, it involves the distribution of energy within that volume. These energies may be in various forms, be it radiant energies from mater sources, or heat energies etc.

Mathematically this energy would be an integral of the energy density over a volume element. In curved spacetime the volume element is factored or weighted by the determinant of the metric tensor corresponding to the given geometry which the spacetime continuum is currently in, encoding geometric properties. The relationship between the volume element and the determinant of metric tensor is useful for describing the geometry of curved spacetime for that given volume of spacetime. This energy is given as;

E = ε_{0} \int d v .

ε_{0} i s t h e e n e r g y d e n s i t y w h e r e \int d v = \int \sqrt{- g} d^{4} x l e a d i n g t o E = \int \sqrt{- g} ε_{0} d^{4} x

Mass density similar to the energy density, characterizes the density of matter content in a given volume of spacetime. This provides insight on the distribution of matter in that region of spacetime. Just like with energy densities, the integrals are also used to calculate total quantity of mass over given volume of curved spacetime.

For the mass density in curved volume of spacetime we define the following;

m = \int \sqrt{- g} ρ d^{4} x, w i t h ρ b e i n g t h e m a s s d e n s i t y

The mass density and energy density contributes to the stress energy on spacetime leading to the curvature of spacetime around that region or volume. The energy content in a region or volume of space may be of various forms coming from astronomical sources and bodies in space.

One can incorporate the mass density to quantum mechanical systems by introducing them as additional terms to the Hamiltonian, and the Schrodinger’s equation. This is analogous to the Schrodinger-Newton equation, where the effect of gravity is being considered in the framework.

Schrodinger-Newton Equation

The Schrodinger-Newton equation [3,4,5] being of the form;

i ℏ \frac{\partial ψ}{\partial t} = (- \frac{i ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x) + m ϕ) ψ

Is a modification of the Schrodinger’s equation [6,7,8] in terms of Newtonian gravitational potential [17,18,19], where a gravitational term is added to describe the interaction of a particle with the gravitational field, it is sometimes referred to as the Schrodinger-Poisson equation.

The Schrodinger-Newton equation assumes that matter waves and particles remain quantum mechanical and gravity remain classical, leading to the equation being considered a semi classical equation.

If the potential term were to be replaced with the solution to Poisson’s equation the schrodinger-newton equation takes the following form;

i ℏ \frac{\partial ψ}{\partial t} = (- \frac{i ℏ^{2}}{2 m} \nabla^{2} + V (x) - G ρ^{2} \int \frac{1}{| X - X_{0} |} d^{3} X) ψ

The equation as stated earlier are semi-classical in nature, relating classical gravity, to quantum matter.

The gravitational potential

ϕ

incorporates the effect of gravity specifically Newtonian gravity into the Schrodinger’s equation, but it does not account for the curvature of spacetime as defined by general relativity [20,21].

However we can rewrite this in terms of the mass density and the volume-element

m = \int \sqrt{- g} ρ d^{4} X

Thus the Schrodinger Newton equation becomes;

i ℏ \frac{\partial ψ}{\partial t} = (- \frac{i ℏ^{2}}{2 m} \nabla^{2} + V (x) + G ρ^{2} \int \sqrt{- g} \frac{1}{| X - X_{0} |} d^{4} X) ψ

This brings in the concept of general relativity through the Integral over curved spacetime volume.

Energy Densities

The energy in a curved volume of spacetime from a quantum mechanical matter source can be defined using elements of quantum field theory, in such case it should be stated that the energy density we’re working with is quantum mechanical and not classical. Defined by the quantum stress energy tensor

ε_{0} = 〈T_{μ v}〉

, where the components of the quantum stress energy tensor is given in terms of the variational derivative of action functional of some assumed quantum field in respect to the metric tensor

g^{μ v}

. Thus the energy is given;

H = E = \int ε_{0} d^{4} X, ε_{0} = 〈T_{μ v}〉 〈T_{μ v}〉 = \frac{- 2}{\sqrt{- g}} \frac{δ S}{δ g^{μ v}}

with S being the action functional for the an assumed quantum field φ

With this knowledge, the expressions below follows;

H_{μ v} = \int \frac{- 2}{\sqrt{- g}} \frac{δ S}{δ g^{μ v}} d^{4} X

We can see that the resulting Hamiltonian is reminiscent of the QFT in curved spacetime framework

The Cosmological Constant

The cosmological constant sometimes interpreted as the energy density of space or vacuum, is the energy density associated with the expansion of the universe and dark energy.

One can also include the cosmological constant to the framework, accounting for dark energy and the universal expansion. To do this the Hamiltonian is then defined as;

H_{μ v} = \int (\frac{- 2}{\sqrt{- g}} \frac{δ S}{δ g^{μ v}} - λ) d^{4} X

where lamba (λ) is the cosmological constant.

This Hamiltonian corresponds to a curved volume of spacetime with a quantum mechanical matter source influenced by universal expansion.

In quantum field theory empty space or vacuum is said to have underlying quantum fields, which can have fluctuations in their ground states at the lowest energy sometimes called zero point energy [11,12,13] which is present at all regions of space-time, it is theorized that the vacuum fields contributes to the cosmological constant. While the Zero point energy from quantum field theory is said to contribute to the energy defined for the cosmological constant, the observed value of the cosmological constant is yet so small compared to the zero point energy said to be contributing to it. This is the cosmological constant problem [14,15,16].

Quantum field theory studies the vacuum energy in terms of zero point energy, while in astrophysics and cosmology the vacuum energy is defined in terms of the cosmological constant. From a possibly unifying perspective the notion that the zero point energy contributes to the cosmological constant’s value of energy arises for this it became an expectation that the value of the vacuum energy with the contributions of zero point energy can be known from the observed cosmological vacuum energy. But this contribution seems annulled in observation because the observed vacuum energy from cosmology appears to be too small to have contributions from the greater value of zero point energy.

Nuclear Energies in Volume of Spacetime

For a nuclear reactions the mass energy equivalence equation

E = m c^{2}

is necessary for calculating the energy of that reaction, however if the mass density of the nuclear system in a volume of curved spacetime is specified, then the mass is given to be

m = \int \sqrt{- g} ρ d^{4} X

, and the equation becomes;

E = \int \sqrt{- g} (ρ c^{2}) d^{4} X

For nuclear fission and radioactive decay processes it can be given as;

E = \int \sqrt{- g} (∆ ρ c^{2}) d^{4} X

This way we get to relate geometry of spacetime to the concept through the determinant of the metric tensor to the energy of the nuclear system.

Negative Energy

Given the energy content in a volume of curved spacetime

E = \int \sqrt{- g} (ε_{0} - λ) d^{4} X

, we can consider setting

ε_{0} \to 0

this gives the equation in flat space-time;

E = \int (- λ) d^{4} X

It is seen that it leads to a negative valued energy [9,10] which drives inflations or expansion of the universe. It is the presence of positive energies, from matter sources and radiations that counteracts this inflationary negative energy.

Conclusions

The Schrodinger-Newton equation has briefly been discussed as a semi classical equation relating Gravity from Newtonian theory of gravity to quantum mechanics, describing how particle behaves in the presence of its self-gravitating field. Gravity is treated classically and matter is treated quantum mechanically, with gravity introduced through an additional term

m ϕ

to the Schrodinger’s equation with

ϕ

being the gravitational potential of the gravitational field.

By re-expressing the mass in terms of the mass density and the integral of the volume element in curved spacetime, the Schrodinger-newton equation for a self-gravitating particle should account for curvature of space-time through the determinant of the metric tensor in the integral of the volume element.

An energy equation is also introduced in terms of energy densities which corresponds to the quantum stress energy tensor in curved space-time such that a part of QFT in curved spacetime arises with the inclusion of the cosmological constant as the vacuum energy density. Nuclear reactions and negative energies were discussed in terms of mass densities in curved spacetime and vacuum energy densities.

With the formulation being presented where the Schrodinger-Newton equation includes elements of general relativity, and the energy in a curved volume of spacetime from quantum matter source also defined using the quantum stress energy tensor and other areas discussed, it may be safe to assume that formulating quantum or semi-classical theories in terms of densities makes it easier to introduce general relativity by means of the volume element in curved spacetime and may be a useful complements to the current framework of Quantum field theory in curved spacetime.

Declarations

I hereby declare that this article, titled; “Quantum densities in Curved Spacetime”.

Is written with no conflicting interest, neither is there any existing or pre-existing affiliation with any institution.
No prior funds is received by the author from any organization, individual or institution.
The content of the article is written with respect to ethics.
The content of the article does not involve experimentation with human and/or animal subjects.
Data-availability; no data, table or software prepared by an external body or institution is directly applicable to this article.

References

Jose W Maluf, Localization of energy in general relativity, Journal of Mathematical Physics, 36(8), 4242-4247, (1995). [CrossRef]
Theodore Frankel et-al, Energy density and spatial curvature in general relativity, Journal of Mathematical Physics, 22(4), 813-817, (1981). [CrossRef]
Richard Harrison et-al, a numerical study of the Schrodinger-Newton equations, Nonlinearity, 16(1), 101, (2002). [CrossRef]
Giovanni Manfredi, The Schrodinger-Newton equation beyond Newton, General relativity and Gravitation, 47(2), 1, (2015). [CrossRef]
Mohammad Bahrami et-al, The Schrodinger-Newton equation and its foundations, New Journal of Physics, 16(11), 115007, (2014). [CrossRef]
Nouredine Zettili, Quantum mechanics: Concepts and applications, John Wiley & Sons, (2009). [CrossRef]
Barry Simon, Schrodinger Operators in the twentieth century, Journal of Mathematical Physics, 41(6), 3523-3555, (2000). [CrossRef]
Feliks Aleksandrovich Berezin et-al, the Schrodinger Equation, Springer Science & Business Media, (2012).
Robert J Nemiroff et-al, an exposition on Friedman Cosmology with Negative energy densities, Journal of cosmology and astroparticle physics, 2015(06), 006, (2015). [CrossRef]
Tomislav Prokopec, Negative energy cosmology and the cosmological constant, arXiv preprint, (2011). arXiv:1105.0078. [CrossRef]
William Simpson & Ulf Leonhardt, Forces of the quantum vacuum: an introduction to Casmir physics, world scientific, (2015).
Simon-Saunders, Is the zero-point energy real? Ontological aspects of quantum field theory, 313-343, (2002). [CrossRef]
Kimball A Milton, The Casmir effect: Physical manifestation of Zero-point energy, world scientific, (2001).
Thomas Banks, The cosmological constant problem, Physics today, 57(3), 46-51, (2004).
Lucas Lombriser, on the cosmological problem, Physics letters B, 797, 134804, (2019). [CrossRef]
Steven Weinberg, the cosmological constant problem, Review of modern physics, 61(1), 1, (1989). [CrossRef]
Eric Poisson & Clifford M Will, Gravity, Gravity, (2014).
Martin Counthan, Presenting Newtonian gravity, European journal of physics, 28(6), 1189, (2007).
RA Capuzzo Dolcetta, Classical Newtonian gravity, Springer, berlin, (2019).
Michael Paul Hobson et-al, General relativity: an introduction for Physicists, Cambridge University Press, (2006).
Robert M Wald, General relativity, University of Chicago press, (2010).

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