Deriving Einstein’s Gravity Equations From Thermodynamics

Emergent gravity is an idea in quantum gravity according to which the fabric of spacetime is not fundamental but emerges as a coarse-graining approximation of underlying (still unknown) microscopic degrees of freedom (similarly to a gas emerging from a large sampling of atoms or molecules). In the words of Huggett and Wuthrich, emergent gravity is the view that gravity arises due to the “collective action of the dynamics of more fundamental non-gravitational degrees of freedom.”

Figure 1: According to emergent gravity, the spacetime continuum can be viewed as the macroscopic limit of some fundamental microscopic structure (source). Here we will investigate a proposal by the physicist Ted Jacobson that Einstein’s gravity can be derived from thermodynamics (source).

In the present article, we will investigate a 1995 proposal by the American theoretical physicist Ted Jacobson that Einstein’s gravity equations can be derived from thermodynamics, “the branch of physics that deals with the relations between heat and other forms of energy” (see link). This implies that Einstein’s equations can be viewed as an equation of state, a thermodynamic equation relating variables describing the state of matter (such as, for example, the ideal gas law).

Figure 2: The French physicist Sadi Carnot considered the “father” of thermodynamics (source), and his 1824 book “Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power,” considered the founding work of thermodynamics (source).

Motivation: Entropy and Horizons in Spacetime

In the 70s, the Mexican-born Israeli-American theoretical physicist Jacob Bekenstein and the English theoretical physicist and cosmologist Stephen Hawking showed that black holes have a thermodynamic entropy proportional to the area of their event horizon.

Equation 1: The Bekenstein-Hawking entropy.

where G, c, h, and k denote Newton’s gravity constant, the speed of light, Planck’s constant, and the Boltzmann’s constant. Note that just by examining the constants in this expression, we infer that black holes lie at the intersection of gravity, quantum mechanics, and thermodynamics since:

• G is the gravitational constant, needed to calculate the gravitational effects in Newton’s law of universal gravitation and in Einstein’s theory of gravity.
• h is the Planck’s constant, the quantum of electromagnetic action relating a photon’s energy to its frequency (photons are particles of light).
• k is the Boltzmann constant, the proportionality factor relating the average kinetic energy of particles in a gas with its thermodynamic temperature.

Figure 3: The Bekenstein-Hawking entropy is the entropy ascribed to black holes. It is proportional to 1/4 of its horizon area (source).

Black holes are not the only spacetime configurations carrying entropy. Two other important examples are:

• Cosmological horizons in de Sitter (dS) space
• Observer-dependent horizons in Rindler spacetime (a coordinate system representing part of Minkowski spacetime that describes uniformly accelerated observers). This type of horizon, which will be central here, has entropy and a temperature (the Unruh temperature). The latter is proportional to the observers’ acceleration and its existence hints thatspacetime itself encodes thermodynamical information.

Figure 4: Jacob Bekenstein (source) and Stephen Hawking (source), the first who showed that black holes have a thermodynamic entropy proportional to the area of their event horizon.

Quoting the renowned American theoretical physicist Robert Wald, who made fundamental contributions to the study of gravitation physics, such as the discovery of the general formula for black hole entropy, and the development of a rigorous formulation of quantum field theory in curved spacetime:

“I believe that the relationship between black holes and thermodynamics provides us with the deepest insights that we currenly have concerning the nature of gravitation, thermodynamics, and quantum physics.”

— Robert Wald

We conclude that, since we can associate a temperature and entropy to regions of spacetime, it is not unreasonable to suppose that their properties may have some similarities to the properties of matter at the macroscopic scale. Quoting Jacobson:

“This perspective suggests that it may be no more appropriate to […] quantize the Einstein equation than it would be to quantize the wave equation for sound in air.”

Ted Jacobson and Einstein’s Equation of State

As described in the introduction, in a 1995 article, Ted Jacobson showed that Einstein’s equations can be obtained by applying thermodynamics laws to the so-called Rindler horizons. His proposal implies that spacetime is emergent.

This idea will be described in detail below, but to fully understand it we first need to review some preliminary concepts from Einstein’s relativity. One of my past articles may be useful as a revision of this material.

Quantum Gravity, Timelessness and Complex NumbersIs The Wave Equation of the Universe Timeless and Real?towardsdatascience.com

It should be noted that if we derive the Einstein equations using arguments from thermodynamics, we cannot interpret the equations geometrically (as it is usually done). If we assume gravity is, by nature, a thermodynamic phenomenon, the Einstein equations must be interpreted using thermodynamic concepts. In other words, gravitational dynamics must be re-expressed in terms of the thermalevolution of spacetime (see this reference).

Preliminary Concepts

We first need to define what are vectors and dual vectors. A manifold can be loosely defined as a space that resembles Euclidean (flat) space near each of its points.

Figure 5: A two-dimensional manifold M with a curve γ parametrized by λ (source).

Consider now a curve γ on the manifold parameterized by λ (see Fig. 5). It can be described by the parametric equations:

Equation 2: Coordinates of the curve γ on M, parametrized by λ.

Fig. 6, shows an example of a curve along the surface of a cylinder. It is given by the following parametric equations:

Equation 3: The λ-parametrized equations of the coordinates of a curve that spirals along the surface of a cylinder. Figure 6: The curve given by Eq. 3.

Now consider a function f defined along the curve γ on M. How does it vary along the curve in terms of the parameter λ?

Equation 4: Variation of the function f along the curve γ in terms of the parameter λ.

where we can identify the components of the tangent vector and the gradient of the function f. The gradient is referred to as a dual vector. Using the “,” notation for partial derivatives we write:

Equation 5: The components of a vector and a dual vector.

Note that in Eq. 4 the Einstein’s summation convention was used to omit the summation symbol.

The figure below shows a tangent vector u to the curve γ parametrized by λ:

Figure 7: The plane tangent to a manifold and a tangent vector at P (source).

Vector and dual vectors transform differently under coordinate transformations:

Equation 6: How the dual vectors and tangent vectors change under a coordinate transformation.

Tensors

We will now consider tensors (see Dirac, for a simple explanation). Vectors are tensors of the type (1, 0). Dual vectors are tensors of the type (0, 1). Following Dirac, to obtain general tensors of higher ranks, we first build the quantity

Equation 7: A simple example of a (2,0)-type tensor.

which is a particular kind of tensor of type (2, 0). Adding several tensors like T, one gets a general tensor of type (2, 0):

Equation 8: An example of a general (2,0)-type tensor.

Under a coordinate transformation, T transforms as:

Equation 9: How a contravariant tensor of second rank changes after a coordinate transformation.

If we have 2 lower (instead of upper) indexes, T is said to be of the type (0, 2). Tensors can also have mixed indexes such as the (1, 1) tensor below:

Equation 10: How the components of mixed tensor change after a coordinate transformation.

Lie derivatives

The Lie derivative is a concept from differential geometry, a mathematical discipline that applies calculus, linear algebra, and multilinear algebra to geometry problems. This type of differentiation, named after the Norwegian mathematician Sophus Lie, evaluates the change of a tensor field along the flowof another vector field (see Wiki).

Figure 8: The Norwegian mathematician Marius Sophus Lie and the front page of his most important treatise, “Theorie der Transformationsgruppen.”

Suppose we have a vector field A in a region of spacetime and a curve γ in the neighborhood of which A is defined. The tangent vector to γ is u = dx/dλ. We consider two points in the curve, x and x+dx.

Figure 9: A vector field A in a region of spacetime, two points x and x+dx inside that region, a curve γ containing both points, and a vector tangent to γ.

Under the infinitesimal change dx

Equation 11: Infinitesimal change of coordinates (see Fig. 9).

the vector A transforms in the following manner:

Equation 12: How the vector A transforms under the change of coordinates in Eq. 11.

Now, the value of the original vector field in x+dx can be written as:

Equation 13: Expansion of the α-component of A(x+dx).

The Lie derivative of A along the curve γ is defined by:

Equation 14: Definition of the Lie derivative of the dual vector A along the curve γ.

The tangent vector u is the direction in which we carry the Lie derivative (see this video for a detailed explanation of the construction of Lie derivatives).

We can better understand the Lie derivative with the following construction. Any vector field is defined by the congruence of curves for which it is the tangent field. We then draw a curve tangent to A at P (the form of the rest of the curve can be anything). We then parametrize the first curve by λ. We choose λ=1 at Pand use the cross curve to fix the parameterization of the other curves to be λ=1. The rate at which λ changes at each curve is fixed by its tangent vectors. We then slide the cross curve by dλ on all curves of the congruence. Note that the cross curve through Q is rigidly tied to the curve through P. There is a vector A’ tangent to the new curve at Q. Since we have two vectors at Q we can subtract them. The Lie derivative is then:

Equation 15: Geometrical definition of the Lie derivative of the dual vector A.

The construction is illustrated below:

Figure 10: The geometrical construction described above.

The Lie derivative is, contrary to its appearance, a tensorial expression since we can rewrite it as:

Equation 16: The Lie derivative written as a tensorial object.

Dual-vectors transform as:

Equation 17: The Lie derivative of the dual vector A along the curve γ.

Now consider a vector A which does not depend on some coordinate, say, x⁰ in some specific coordinate system:

Equation 18: A does not depend on x⁰ in some specific coordinate system.

The * indicates that the corresponding equality is valid in one specific coordinate system. Hence we have a set of curves along which x⁰ increases where A does not change. In this coordinate system:

Equation 19: Tangent vector to set of curves along which x⁰ increases where A does not change Figure 11: Set of curves along which x⁰ increases where A does not change.

From Eq. 19 we obtain:

We can rewrite Eq. 18 as:

These equations imply that:

Equation 20: The Lie derivative of the vector A along the curve with tangent vector U.

This equation expresses the invariance of A in the direction of U. But the Lie derivative on the left-hand side is independent of coordinate systems. Therefore, since it is zero in one coordinate system, it is zero in all coordinate systems.

The Lie derivative of a type-(0,2) tensor is:

Equation 21: The Lie derivative of a type-(0,2) tensor.

Consider some coordinate system the tensor A is independent of some coordinate x⁰. These can be expressed in two different ways:

• To say this in terms of coordinates is Eq. 18
• To say this covariantly is to say that the Lie derivative is zero when the vector field U is aligned with the coordinates where x⁰ is running.

Equation 22: The way to expressing covariantly that A is invariant as we make a translation in a certain direction (in this case U) in spacetime.

Eq. 22 expresses the invariance of the tensor A in the direction of the vector U.

Figure 5: Two ways to express tensor that A is independent of some coordinate x⁰.

This notion becomes very important if we have a symmetry in our spacetime. For example, if vectors don’t depend on time x⁰ or if vectors don’t depend on rotations along some axis, the way to express this in a covariant way is to say that the Lie derivative of that tensor is going to be zero along the appropriate direction, either at time translation or a rotation along an axis. The vector along which you have the symmetry becomes the so-called Killing vector.

Figure 12: The German mathematician Wilhelm Killing (source) and the front page of his article “Die Zusammensetzung der stetigen endlichen Transformationsgruppen” (Math. Ann.33, 1–48) which was considered by the renowned Canadian mathematician A. J. Coleman to be “the most significant mathematical paper he has read or heard about in fifty years” (source).

Killing Vectors and Symmetries

A Killing vector is a vector field ξ such that the Lie derivative of the metric along ξ is zero:

Equation 23: Definition of a Killing vector.

If in a given coordinate system the metric does not depend on the coordinate σ*, the α-component of ξ is:

We can also write ξ and its α-component as:

Eq. 23 says that we have a symmetry of the metric in the direction along which you know ξ points. Symmetries of the metric are called isometries.

Using the definition of Lie derivative we obtain:

Equation 24: Condition obeyed by the Killing vector.

We can use also Killing vectors to obtain constants of motion along geodesics. If u is tangent to a geodesic it is trivial to show that for ξ obeying Eq. 23, the following result follows:

Equation 25: This equation, which is can be easily proved using the equations above, gives us constants of motion along geodesics.

Isometries give origin to conserved quantities along geodesics. More precisely, Killing vectors ξ generate isometries, and transformations under which g is invariant are expressed infinitesimally as motions in the direction of ξ.

For clarity, let us now consider some simple examples of Killing vectors and their corresponding symmetries.

Example 1
Take for example the following metric in R³:

Equation 26: Example of metric in R³.

Note that the metric does not depend on x, y, or z. Therefore, the following three vectors are Killing vectors corresponding to translations:

Equation 27: Three Killing vectors associated with the metric in Eq. 26.

There are other symmetries in R³. Writing Eq. 26 in spherical coordinates, (illustrated below), the metric becomes:

Equation 28: Metric in Eq. 26 expressed in spherical coordinates. Figure 13: Definitions of the variables in Eq. 28.

Since the components of g are ϕ-independent,

Equation 29: The Killing vector corresponding to the ϕ-independence of the metric.

is another Killing vector of R³. In cartesian coordinates this becomes:

Equation 30: The Killing vector in Eq. 29 written in cartesian coordinates.

Rotations around the other two axes give us two other Killing vectors.

Example 2
Consider now a spherically symmetric spacetime (such as the Schwarzchild metric):

Equation 31: Spherically symmetric spacetime.

Since the metric g has no dependence on t and no dependence on ϕ, time translations, and rotations around the z-axis are examples of isometries. We have two obvious Killing vectors:

Equation 32: Two Killing vectors associated with metric Eq. 31.

The following quantities (related to energy and angular momentum per unit mass) are constant along geodesics to which a vector u is tangent:

Equation 33: Two constants along geodesics to which u is tangent.

Killing Horizons

Let us consider a null hypersurface (for example, a light cone). It is, by definition, a hypersurface whose normal vector at every point is null (it has zero length with respect to the local metric tensor g). A Killing horizon Σ is a null hypersurface where the norm of a Killing vector field vanishes. Also, since a null surface cannot have two linearly independent null tangent vectors, ξ will be normal to Σ.

In a Minkowski spacetime in inertial coordinates, the timelike Killing vector that generates boosts, for example, in the x-direction is given by:

Equation 34: The Killing vector that generates boosts in the x-direction in a Minkowski spacetime.

Its norm is given by:

Equation 35: The norm of the Killing vector given by Eq. 34.

When ξ is constant its orbits are hyperboles representing worldlines of uniformly accelerated observers with proper acceleration a = 1/ξ. As ξ → 0 the acceleration increases and the boost Killing vector field generates a bifurcate Killing horizon,

Equation 36: The null surfaces corresponding to Eq. 35.

the so-called Rindler horizon.

These null surfaces are therefore Killing horizons. Since a Killing vector ξ is normal to its Killing horizon Σ, along Σ it obeys the geodesic equation (the κ on the right-hand side of the geodesic equation accounts for the possibility that the integral curves of ξ are not affinely parametrized):

Equation 37: Definition of the surface gravity κ. Note that κ must be evaluated at Σ.

where κ is called surface gravity. For a static spacetime, κ is the acceleration of a static observer close to the horizon, measured by a static observer at ∞.

The Rindler Wedge

Consider an arbitrary spacetime point p. Locally, the spacetime around p is flat (because of the principle of equivalence). Now choose a small patch B of a spacelike 2-surface passing containing P and introduce Riemann normal coordinates (RNC). The metric in RNC is given by:

Equation 38: Metric in RNC at p.

The coordinates of the points on the patch B are:

Equation 39: Coordinates of the points on the patch B.

Adding past and future light sheets, say, in the z-direction we obtain:

Equation 40: Coordinates on the past and future light sheets in the z-direction.

Eq. 40 describes a local Rindler wedge (two null three-dimensional half-planes joined by the spacelike bifurcation, the 2-plane B at t=0) illustrated below.

Figure 14: The Rindler wedge (source).

Consider now a sheet of hyperbolic timelike observers close to the null surface. Their coordinates, velocity, and acceleration

Equation 41: Coordinates, velocity, and acceleration of hyperbolic timelike observers close to the null surface.

where a is the observer acceleration

describe approximately uniformly accelerated observers. For such observers, the light sheets of the Rindler wedge form the Rindler horizon. A causal horizon is the boundary of the spatial region consisting of points causally connected to an observer. More precisely, a causal horizon is a hypersurface that is a boundary between light rays that are directed outwards and moving outwards (dz/dt>0), and those directed outward but moving inward (dz/dt