Deriving Einstein’s Gravity Equations From Thermodynamics

Is Gravity Just an Average of the Behavior of Unknown “Atoms” of Spacetime?

Image for post
Image for post
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).
Image for post
Image for post
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

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

“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

“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

Preliminary Concepts

Vectors and Dual Vectors

Image for post
Image for post
Figure 5: A two-dimensional manifold M with a curve γ parametrized by λ (source).
Image for post
Image for post
Equation 2: Coordinates of the curve γ on M, parametrized by λ.
Image for post
Image for post
Equation 3: The λ-parametrized equations of the coordinates of a curve that spirals along the surface of a cylinder.
Image for post
Image for post
Figure 6: The curve given by Eq. 3.
Image for post
Image for post
Equation 4: Variation of the function f along the curve γ in terms of the parameter λ.
Image for post
Image for post
Equation 5: The components of a vector and a dual vector.
Image for post
Image for post
Figure 7: The plane tangent to a manifold and a tangent vector at P (source).
Image for post
Image for post
Equation 6: How the dual vectors and tangent vectors change under a coordinate transformation.

Tensors

Image for post
Image for post
Equation 7: A simple example of a (2,0)-type tensor.
Image for post
Image for post
Equation 8: An example of a general (2,0)-type tensor.
Image for post
Image for post
Equation 9: How a contravariant tensor of second rank changes after a coordinate transformation.
Image for post
Image for post
Equation 10: How the components of mixed tensor changes after a coordinate transformation.

Lie derivatives

Image for post
Image for post
Figure 8: The Norwegian mathematician Marius Sophus Lie and the front page of his most important treatise, “Theorie der Transformationsgruppen.”
Image for post
Image for post
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.
Image for post
Image for post
Equation 11: Infinitesimal change of coordinates (see Fig. 9).
Image for post
Image for post
Equation 12: How the vector A transforms under the change of coordinates in Eq. 11.
Image for post
Image for post
Equation 13: Expansion of the α-component of A(x+dx).
Image for post
Image for post
Equation 14: Definition of the Lie derivative of the dual vector A along the curve γ.
Image for post
Image for post
Equation 15: Geometrical definition of the Lie derivative of the dual vector A.
Image for post
Image for post
Figure 10: The geometrical construction described above.
Image for post
Image for post
Equation 16: The Lie derivative written as a tensorial object.
Image for post
Image for post
Equation 17: The Lie derivative of the dual vector A along the curve γ.
Image for post
Image for post
Equation 18: A does not depend on x⁰ in some specific coordinate system.
Image for post
Image for post
Equation 19: Tangent vector to set of curves along which x⁰ increases where A does not change
Image for post
Image for post
Figure 11: Set of curves along which x⁰ increases where A does not change.
Image for post
Image for post
Image for post
Image for post
Image for post
Image for post
Equation 20: The Lie derivative of the vector A along the curve with tangent vector U.
Image for post
Image for post
Equation 21: The Lie derivative of a type-(0,2) tensor.
Image for post
Image for post
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.
Image for post
Image for post
Figure 5: Two ways to express tensor that A is independent of some coordinate x⁰.
Image for post
Image for post
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

Image for post
Image for post
Equation 23: Definition of a Killing vector.
Image for post
Image for post
Image for post
Image for post
Image for post
Image for post
Equation 24: Condition obeyed by the Killing vector.
Image for post
Image for post
Equation 25: This equation, which is can be easily proved using the equations above, gives us constants of motion along geodesics.
Image for post
Image for post
Equation 26: Example of metric in R³.
Image for post
Image for post
Equation 27: Three Killing vectors associated with the metric in Eq. 26.
Image for post
Image for post
Equation 28: Metric in Eq. 26 expressed in spherical coordinates.
Image for post
Image for post
Figure 13: Definitions of the variables in Eq. 28.
Image for post
Image for post
Equation 29: The Killing vector corresponding to the ϕ-independence of the metric.
Image for post
Image for post
Equation 30: The Killing vector in Eq. 29 written in cartesian coordinates.
Image for post
Image for post
Equation 31: Spherically symmetric spacetime.
Image for post
Image for post
Equation 32: Two Killing vectors associated with metric Eq. 31.
Image for post
Image for post
Equation 33: Two constants along geodesics to which u is tangent.

Killing Horizons

Image for post
Image for post
Equation 34: The Killing vector that generates boosts in the x-direction in a Minkowski spacetime.
Image for post
Image for post
Equation 35: The norm of the Killing vector given by Eq. 34.
Image for post
Image for post
Equation 36: The null surfaces corresponding to Eq. 35.
Image for post
Image for post
Equation 37: Definition of the surface gravity κ. Note that κ must be evaluated at Σ.

The Rindler Wedge

Image for post
Image for post
Equation 38: Metric in RNC at p.
Image for post
Image for post
Equation 39: Coordinates of the points on the patch B.
Image for post
Image for post
Equation 40: Coordinates on the past and future light sheets in the z-direction.
Image for post
Image for post
Figure 14: The Rindler wedge (source).
Image for post
Image for post
Equation 41: Coordinates, velocity, and acceleration of hyperbolic timelike observers close to the null surface.
Image for post
Image for post
Image for post
Image for post
Equation 42: The Unruh temperature.

Deriving The Equation of State

Geometrical Construction

Image for post
Image for post
Figure 15: A spacelike foliation parametrized by a time coordinate t
Image for post
Image for post
Equation 43: The tangent vector along the congruence increasing towards the future.
Image for post
Image for post
Equation 44: Choice of the Killing vector ξ.
Image for post
Image for post
Equation 45: The surface element for the local Rindler horizon where dA is the codimension-two spacelike cross-sectional area element.
Image for post
Image for post
Figure 16: Illustration of the geometric construction described in the text.
Image for post
Image for post
Equation 46: Heat flux to the past of B.
Image for post
Image for post
Equation 47: The area variation of H as δQ crosses it.
Image for post
Image for post
Equation 48: The area variation of the generators of the horizon.
Image for post
Image for post
Equation 49: Clausius relation.
Image for post
Image for post
Equation 50: The entropy S is proportional to the area of the horizon A.
Image for post
Image for post
Equation 51: The evolution of the congruence of null geodesics generating the horizon is described by the Raychaudhuri equation.
Image for post
Image for post
Equation 52: The entropy variation.
Image for post
Image for post
Image for post
Image for post
Equation 53: Einstein’s equation of motion as an equation of state.

Data Scientist | Physicist | Lover of unification, generalization & abstraction | https://www.linkedin.com/in/marco-tavora

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store