Understanding the Relationship Between Time and Spatial Geometry
Written on
Chapter 1: The Foundations of Geometrodynamics
The essence of Albert Einstein's initial formulation of general relativity, which serves as his theory of gravity, revolves around the metric tensor ( g ), with the theory being covariant in spacetime. Covariance, more specifically general covariance, refers to the invariance of physical laws under any arbitrary coordinate transformations. The premise is that since coordinates are merely human-constructed labels, the laws of physics should remain unaffected by the manner in which they are designated.
In the framework of general relativity, the action—an attribute that encapsulates a system's dynamics from which its motion equations can be derived—is identified as the Einstein–Hilbert action:
S = int d^4x sqrt{-g} left( frac{R(g)}{16 pi G} + Lambda right)
where ( g = text{det}(g) ), ( R(g) ) represents the Ricci scalar (a measure of how the volume of a small ball in a curved manifold contrasts with that in Euclidean space), ( G ) is Newton's constant, and ( Lambda ) is the cosmological constant (which will be omitted for simplicity). Here, ( S ) should include a boundary integral if the spacetime in question is bounded. For our discussion, we will assume no matter is present.
When we set the variation of this action concerning the (inverse) metric to zero, we derive Einstein's source-free field equations:
R_{munu} = 0
where ( R ) is the Ricci curvature tensor.
To elucidate these concepts, we can draw an analogy with particle mechanics. In Lagrangian mechanics, we are provided with a Lagrangian ( L ), from which we construct an action functional ( S ) and derive the motion equations by extremizing ( S ):
S = int L dt
This raises the question: How do we understand the connection between time and the geometry of space?
The objective of this article is to illustrate that time is indeed encoded within the spatial geometry. This discussion follows the groundwork laid by Baierlein, Sharp, and Wheeler in their 1962 paper, which I will refer to as BSW. As articulated in the renowned text "Gravitation" by Misner, Thorne, and Wheeler, "three-geometry carries information about time."
This relationship arises from the extension of the Lagrangian methodology to general relativity:
S = int left( text{Intrinsic Geometry}_1 + text{Intrinsic Geometry}_2 + text{4-D Geometry} right) dt
The goal is to identify a four-geometry that aligns with Einstein's equations and reduces to the three-geometries on the surfaces ( sigma ) and ( sigma' ). Quoting "Gravitation," we adjust the four-geometry to "extremize the action" given the three-geometries of the two faces of a "sandwich of spacetime." This approach leads us to:
- Determine the time-like separation between two three-dimensional surfaces, or as Barbour puts it, "how far apart in time" the three spaces are.
- Identify the positions of these surfaces within spacetime.
In this four-dimensional spacetime framework, each three-geometry is defined solely based on the two intrinsic geometries.
Before delving deeper, it’s important for readers to familiarize themselves with the ADM formalism in general relativity, which will be covered in the next section.
Section 1.1: A Glimpse at the ADM Formalism
In the ADM formalism—named after its contributors, Ricard Arnowitt, Stanley Deser, and Charles Misner—general relativity is articulated as a dynamic theory, framed as an initial value problem.
The dynamics within general relativity are termed Geometrodynamics. This section will illustrate that the entities that undergo change in general relativity are distances within three-dimensional surfaces situated in four-dimensional spacetime, rather than distances in four-dimensional spacetime itself.
The configuration space in this dynamic version of general relativity is termed superspace.
The ADM construction can be visualized as follows:
The shift vector ( N ), depicted in the figure, quantifies the "distortion of the surface as it evolves over time." The proper distance between two surfaces is expressed as ( dtau = N_0 dt ), where ( N_0 ) (or ( N )) is the lapse function. This function indicates the rate at which proper time changes concerning the time label of the surfaces ( Sigma(t) ).
Utilizing the illustration above, we can reformulate the distance ( ds^2 ) with the aid of the shift and lapse functions:
ds^2 = g_{munu} dx^mu dx^nu
where ( g ) is the metric of the three-surface. The extrinsic curvature of the three-dimensional hypersurfaces embedded in the four-dimensional spacetime is given by:
K = frac{1}{2} left( nabla_{mu} n_{nu} + nabla_{nu} n_{mu} right)
Here, the symbol ( | ) denotes covariant differentiation concerning the intrinsic spatial metric within the surfaces.
The Ricci scalar ( R ) can be articulated in terms of the extrinsic curvature ( K ), its trace ( K ), and the intrinsic three-curvature ( R^3 ) (the three-dimensional variant of the Ricci scalar):
R = f(K, K, R^3)
This leads us to express the gravitational Lagrangian density as follows:
mathcal{L} = f(R^3, K, K)
The Lagrangian density does not include time derivatives of the shift and lapse functions. Denoting the momenta conjugate to ( N_0 ) and ( N_i ) as ( pi ) and ( dpi/dt ) respectively, we establish that:
pi = 0 quad text{and} quad pi^i = 0
This implies that both ( pi ) and ( pi^i ), representing lapse and shift, are non-dynamical variables (they merely measure surface deformations ( Sigma )). Consequently, we derive conditions on the Hamiltonian ( H ):
H = mathcal{H}(N, N_i)
The tensor ( G ) in the Hamiltonian framework is known as the Wheeler–DeWitt metric.
Ultimately, the equations form a Hamiltonian dynamical system characterized by constraints, offering a reformulation of Einstein's field equations that delineates the evolution of three-metrics:
H = H(text{constraints}) = text{evolution of three-metrics}
Paul Dirac, a pivotal figure in 20th-century physics, alongside John Wheeler, noted the elegant simplicity of the Hamiltonian formulation, which prompted them to reconsider the fundamental nature of spacetime.
Dirac asserted:
> "This result has led me to doubt how fundamental the four-dimensional requirement in physics is."
Wheeler reflected:
> "The dynamic object is not space-time. It is space."
As we explore geometrodynamics using the action derived from our previous equations, we can determine the temporal evolution of a three-geometry of curved empty space by specifying:
- The geometry of the initial surface (the intrinsic curvature)
- The extrinsic curvature of the initial surface ( K )
However, ( h ) and ( K ) cannot be independently specified; they must adhere to the initial value equations established by Foures and Lichnerowicz.
Section 1.2: The BSW Approach to Geometrodynamics
Now, let’s walk through the steps outlined in BSW (the prefactor ( 1/16pi G ) will be disregarded):
Step 1: Choose two very similar three-dimensional metrics:
g_{1} approx g_{2}
The time separation between the surfaces is finite, conveniently set as ( Delta x^0 = 1 ).
Step 2: Introduce a yet undetermined four-geometry filling the space between the surfaces. The separation between two points can be denoted as:
x^mu = (x^0, x^i)
Step 3: A four-geometry that extremizes the BSW action will satisfy Einstein's field equations.
The action integral is given by:
S = int mathcal{L} sqrt{-g} , d^4x
In ADM terms, we have:
S = int mathcal{L}_{text{ADM}} dt
Step 4: Assuming the three-geometries are nearly identical leads to:
V approx V_1 approx V_2
Step 5: By extremizing with respect to ( eta_0 ), we can derive:
eta_0 text{ (proper time separation)}
Step 6: Here, ( eta_0 ) depends on ( kappa ), which is linked to the shift vector ( eta_i(x^1, x^2, x^3) ).
Step 7: Solve for ( eta_i ) (with appropriate boundary conditions) and substitute into the previous equations to determine the time separation ( eta_0 ) in terms of intrinsic geometries.
Step 8: The extrinsic curvature ( K ) can be derived, leading to the conclusion that Einstein's equations, alongside the initial three-metric and extrinsic curvature, determine the four-metric of the embedded spacetime.
BSW demonstrated how to ascertain the time-like separation between the two surfaces and their positioning in spacetime, given the intrinsic geometries of three-dimensional surfaces.
This video, titled "Entanglement and the geometry of spacetime by Matthew Headrick," delves into the intricacies of how entanglement intertwines with the structure of spacetime.
In this insightful video, "How Can SPACE and TIME be part of the SAME THING?", the concepts of space and time are examined, revealing their interconnected nature.
Thank you for reading! I welcome any constructive feedback and suggestions. For more insights on physics, mathematics, and machine learning, feel free to visit my LinkedIn, personal website, or GitHub at www.marcotavora.me!