Chapter 1: Understanding Dark Energy Through Mathematics

The upcoming series of articles takes a deeply mathematical approach to understanding dark energy. If you're not inclined to explore the derivation of equations that test the theories about our universe's origins, you may want to skip ahead to Article 33, which poses the intriguing question: Is our universe situated within a wormhole? The central theory suggests that our universe shares similarities with scenarios arising when a wormhole is generated inside a black hole within Anti-de Sitter (AdS) space. Professor Susskind from Stanford University formulated the foundational equations as part of his ongoing research into computational complexity.

Inside the Realm of Black Holes

Many researchers propose that the outer layer of a black hole functions as a hologram. Susskind sought to articulate the interior of a black hole using holographic principles. He posits that:

When a star collapses, a horizon is established, and its area expands until it attains a stable value. This phenomenon can be observed from outside the horizon and is a manifestation of the second law of thermodynamics: entropy increases until thermal equilibrium is achieved.

However, there exists another related yet lesser-known occurrence: the growth of spatial volume behind the black hole's horizon. Initially, this might seem like another instance of entropy increase, but a more thoughtful examination reveals otherwise. The expansion of the interior continues long after the black hole has reached thermal equilibrium. The nature of this 'something else' that increases remains one of the profound mysteries in black hole physics — unless someone thought to investigate it.

Susskind theorizes that this 'something' is rooted in quantum computational complexity. While this article won't delve into the intricate details of Susskind's derivations, a brief overview of his work (co-authored with Adam Brown) on 'The Second Law of Quantum Complexity' illustrates the mathematical depth required to grasp his arguments:

We propose that there exists a thermodynamic framework for quantum complexity, which encompasses a "second law of complexity." We establish a correspondence between the computational complexity of a quantum system with K qubits and the positional entropy of a corresponding classical system with 2K degrees of freedom. We further assert that the kinetic entropy of the classical system equates to the Kolmogorov complexity of the quantum Hamiltonian. Our observations indicate that the expected growth pattern of quantum system complexity parallels the classical system's entropy growth. We also argue that the concept of 'uncomplexity' serves as a resource that can be utilized for directed quantum computation. While not primarily focused on black holes, we derive a surprising interpretation of the uncomplexity resource as the accessible volume of spacetime behind a black hole's horizon. (Brown and Susskind)

The Brown/Susskind paper posits that the Second Law of Complexity for a quantum system is a natural outcome of the second law of thermodynamics for an auxiliary classical system. Drawing from Michael Nielsen's work on geometrizing complexity, Susskind's mathematical framework for the interior of a black hole assumes an AdS space characterized by constant negative curvature. A hyperbolic plane serves as a representation of this curvature, appearing saddle-shaped in two dimensions, where a single point can represent both a maximum in one dimension and a minimum in another. Visualizing a hyperbolic plane in dimensions beyond two proves challenging.

Susskind's Second Law of Computational Complexity

Susskind's Second Law of Computational Complexity suggests that the equations he derives for complexity within an AdS space (the bulk) have complementary interpretations regarding the boundary of that space. He contends that computational complexity rises linearly up to a maximum over time.

Estimating the Universe's Expansion Rate

According to Einstein, gravity is woven into the fabric of space, which exists along the boundary of an AdS bulk space. One possible way to manifest the growth in quantum complexity is by establishing a density that represents the fabric's capacity to store information, expressed as [V? * T?], where V? indicates the volume of space at time t and T? represents the temperature at that time. Temperature is included in this equation because, as Professor Lloyd from MIT notes, the universe's computational capacity is dependent on the temperature of space.

By integrating Susskind's and Lloyd's insights, we arrive at the equation:

C? ~ [V? * T?] = a * t

Where:

• C? = Quantum computational complexity at time t.
• V? = Volume of the universe at time t (currently estimated at 1.2151 * 10^13 mega-parsecs).
• T? = Temperature of the universe at time t (currently 2.73 ?K).
• a = Rate of complexity increase.
• t = Age of the universe at time t (currently 13.8 billion years).

To quantify the rate of growth in the universe's computational capacity (dV/dt):

V * T = a * t

log (V) * log (T) = log (a) + log (t)

To determine the change in the universe's temperature, we can hypothesize that dark energy contributes information to the universe. Cosmologists believe that the universe's temperature has gradually decreased over time. As new information is incorporated, the universe's average temperature diminishes. The 'temperature' associated with this new information can be computed based on the decline in average temperature. Physicists have estimated the universe's temperature to be approximately 5.1 ?K, six billion years after the Big Bang, compared to the current temperature of 2.73 ?K.

The expansion of the universe's size is contingent on the cosmological model applied. A graph illustrating the universe's size over time suggests it may have doubled in size about 6.375 billion years post-Big Bang (approximately 7.425 billion years ago) at a relatively consistent rate. Assuming a linear growth rate, as complexity is also believed to increase linearly, we can express the growth with the equation:

Increase in size of universe = — 0.71717 + (1/7.425) * increase in age in billion years

This assumes that 6.375 billion years is when the universe was half its current size, a deduction made to calculate the universe's expansion rate, estimated to be around 72 km/sec/megaparsec. The estimates in this article aim to demonstrate consistency with actual observations. From the baseline of 6.6 billion years after the Big Bang (based on physicists' published research on temperature), the universe's volume (V) has increased by a factor of 1.8865. The formula reflecting this information is:

V * T? + V * (0.8865) * T? = V * 1.8865 * T?

Where:

• V = Volume of the observable universe at 6.6 billion years after the Big Bang.
• T? = Temperature of the universe after 6.6 billion years (5.1 ?K).
• T? = Current temperature of the universe (2.73 ?K).
• T? = Temperature increase associated with each new piece of information.

Thus, T? = (1.8865 * 2.73 - 5.1) / 0.8865 = 0.0565 ?K, indicating that each new piece of information has a 'temperature' just above absolute zero. This value may represent the 'temperature' of dark matter, although it's statistically measured. The average temperature change in the universe for a small increment of time or information is the temperature of new information divided by the universe's computational capacity, represented by its age.

The change in average temperature associated with a unit of information (dT/dt) is negative T? / t since T? is lower than the current average universe temperature (2.73 ?K).

Thus, dT/dt = — 0.0565 / (13.8 * 31.536 * 10^15) = — 0.1298 * 10^–18 ?K

And:

d²T/dt² = T²/t² = 0.0565 / (13.8 * 31.536 * 10^15)² = 2.983 * 10^-37 ?K

Earlier calculations yield:

dV/dt = V/t — (V/T) * dT/dt = 2.85 * 10^–5 mps/second = 72 km/sec/mps

Where:

• V = 1.2151 * 10^13 mega-parsecs (mps).
• T = 2.73 ?K.
• t = 13.8 billion years.

Furthermore:

d²V/dt² = (1/t) * dV/dt — V/t² + V * (1/T²) * d²T/dt² — (1/T) * dT/dt * dV/dt

= 0.3 * 10^–23 mps/second = 7.62 * 10^–19 km/sec/mps

These findings indicate that Susskind's equation for Computational Complexity aligns with current estimates of the universe's expansion rate (72 km/second/megaparsec) and the observation that this rate appears to be accelerating. However, the predicted acceleration rate is so minimal that it cannot yet be accurately measured with existing technology.

The central question of this article is:

Is our universe a hologram within a black hole?