In recent years, we have seen the demonstration of profound relationships between the geometry of space-time and thermodynamics or quantum information [1] [2] . In all cases it results in a derivation of Einstein’s equations as a consequence either of a thermodynamic equilibrium condition at a causal horizon, or of counting the number of possible flips in a space-time volume and changing the area of a space-like surface by a suitable multiple of that number. In this paper we are going to combine these approaches to show that Einstein’s equations actually follow from an information consistency condition under a thermal evolution. In addition we will also see that it does not make a difference whether we consider a mass and its gravitational field or a charge and its Coulomb and (induced) magnetic fields. Both sources of energy influence the geometry of space-time by the same mechanism.

There has been recent work done around the concept of thermal time and its relations to other, geometric time-flows [3] -[7] . We will make the thermal flow of a physical system, represented by a density operator, the starting point of our considerations.

Let there be a physical system with density operator and a Hamiltonian H. The cosine of the angle between the initial state and any further state under the Schroedinger evolution is given by the scalar product. A future state is called distinguishable from, if it is orthogonal, i.e.. The minimal time needed for to evolve into an orthogonal state is given by the Margolus-Levitin bound [8] ,

Here denotes the average energy of the system¹ and h denotes the Planck constant. If, for instance, the number of degrees of freedom of the system is large, we may assume that the value is part of the spectrum and hence the minimum-time is actually attained. Since general relativity is a macroscopic theory, this will most likely hold for a system. If, instead of flipping between two orthogonal states, one

considers passing through a long sequence of orthogonal states, then the minimal time turns out to be We will make use of both results.

We now suppose that the system is part of a heat bath at temperature T and consider the flow generated under the assumption that the density operator represents itself thermal equilibrium, i.e. maximum entropy, and the Hamiltonian is therefore, where k denotes the Boltzmann constant. By using this Hamiltonian Equation (1) turns into

In this equation denotes the (von Neumann) entropy of the system. Equation (2) allows us to define a (process) velocity, i.e. the average number of orthogonal states passed by unit of (proper) time

.2 (3)

Expression (3) further allows us to define a very general equilibrium condition between two systems [6] . Given two systems and with their respective temperatures and, and corresponding observers with their proper-time intervals and, we say that two systems are in equilibrium relative to these observers, if there holds

Note that condition (4) can also be viewed as a consistency condition for two sub-systems of an overall system in thermal equilibrium, as indicated in [9] . It is important that with a relation like (4) we actually consider classical information in form of distinguishable states, which appears natural, if a macroscopic theory like relativity is in scope.

Equation (4) is a relational expression, which we now apply to the situation of observers in a four-dimensional space-time. A space-time allows an equilibrium, if it is static [6] [10] , i.e. if its local line-element does not explicitly depend upon the time-coordinate and is therefore of the form

From Equation (5) it is immediately evident that for two observers along their time-like world-lines with and, relation (4) turns into

This is the well-known Tolman-Ehrenfest effect [10] . This elegant derivation already indicates the usefulness of definition (4).

We now consider a situation, where the space-time is actually a flat Minkowski space-time and the observers move with constant acceleration. To generate a static environment we work in the Rindler co-moving coordinate frame and hence in the Rindler-wedge. In Rindler-coordinates the line-element takes the form

For the constant acceleration a there holds.³

For an observer along a time-like world-line with and a system with relations (3) and (4) result in

By a calculation to normalize states, the value chosen for the constant is and Equation (8) turns into the Unruh-relation with corresponding Unruh temperature.⁴

In the last couple of years, we have seen a number of different ways to derive Einstein’s equation by means of the holographic principle and thermodynamic equilibrium assumptions [1] [14] . A flux of energy through a (local) causal horizon causes some kinds of gravitational lensing effect in order to maintain the thermodynamic equilibrium. It seems that the other direction of reasoning, i.e. to derive the holographic principle from Einstein’s equations, is much harder and the principle lies deeply hidden in the structure. Another approach starts from an information-theoretic basis, counts the maximal number of flips in a space-time volume and then carves out a corresponding number of area-elements from a horizon surface to cause it to curve [2] . The two approaches seem little related at first.

In this paper we combine the two perspectives insofar, as we follow the thermodynamic arguments to derive Einstein’s equations from the holographic principle but, instead of assuming it to start with, we derive it by an information theoretic approach. We define a consistency relation across different observers, which observe different systems in thermal equilibrium, by demanding that they all agree on the same maximal number of orthogonal states (flips), which their system can pass in thermal evolution per unit of (proper) time. We apply this definition to observers in a Rindler-frame and if acceleration happens because of a gravitational or an electrostatic force, then the observers identify the available information or entropy with a quotient of the horizon area and some fundamental area units, the Planck or Compton spheres.

In fact expression (10) is very instructive and can be interpreted in the following way: if a falling observer feels an acceleration while observing a hypothetical one bit system, then another observer with acceleration attributes to his system an entropy proportionate to A_R. The only system present in this set-up is space-time itself and (10) and its consequences add to the evidence that entropy is actually attributable to

space-time and has an elementary unit of.

Andreas E.Schlatter, (2015) On the Relationship between the Geometry of Space-Time and Its Information Content. Journal of Modern Physics,06,2184-2190. doi: 10.4236/jmp.2015.615221