In the Proceedings of the Royal Society of London, in 1929, one of the giants of twentieth century scientific community, Paul Dirac narrated at a moment: “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known.” However, the advancements in technology and innovative mathematical theories to study the nature at its fundamentals made this statement quite vague, in essence, with the passage of time. People also stated the same when the discovery of Higgs boson completed the mysterious puzzle of standard model (SM) and made heuristic additions to our understanding of the universe but then cosmology played its role to change the game and showed that the successful SM explains only around four percent of the universe while the nature and dynamics of the rest of the twenty-six percent dark matter and the seventy percent dark energy cannot be explained under its realm as shown in figure 1.1. These and some other compelling arguments again convinced the scientists to think of a more fundamental formalism to describe the underlying phenomena of nature, that is, a theory of quantum gravity – the merger of classical general relativity and quantum mechanics.

Figure 1.1: Pie chart of the content of the universe (Credits: NASA/WMAP Science Team)

What is quantum gravity?

In the last few decades, several approaches including string theory, modified Newtonian dynamics, loop quantum gravity and discrete causal theories were proposed as a solution to the problem of quantum gravity (QG). In the present series, I will try to familiarize the general audience with an important contender of discrete gravity, the causal set approach, one of the versions of discrete causal theory. However, before going into the details of the idea of a discrete theory and its formalism, let us briefly discuss some open issues which demand a quantized version of spacetime geometry.

Since the formulation of Einstein’s general relativity and quantum mechanics, both theories were (and are) successfully tested millions of times which depicts how much important these theories are to modern day science. In the formalism and ideas, both are far apart. General relativity is classical in nature while quantum mechanics upholds the idea of quantization and interference at the subatomic scale. But at the most fundamental level, nature is in the favor of ‘discrete spacetime’. The fundamental scale, at which any theory of QG can hold, is thought to be Planckian (and sub–Planckian) scale which is a combination of three fundamental constants of nature, namely, speed of light (c), Planck’s constant (h) and Newton’s gravitational constant (G). We will see later, in detail, that cosmology is one of the most important natural observatories, to seek quantum gravity.

From the evolution history of universe, it is known that present infrastructure of physics tells us nothing about very early universe or what happens at the sub–Planckian scale. Although, cosmological solutions, by agreeing to the observations, successfully predict an accelerating universe. From this result, it can be extrapolated that this expansion was started, at some zero cosmic time, from a highly dense point (or singularity) with no spatial dimensions. This is where classical GR fails to yield any feasible result(s). This inability to extract any information from this singularity and the subsequent Planck era signifies the presence of quantized theory of gravity.

Black holes also qualify to be a testable region for quantum gravity. In the early seventies, Bekenstein interpreted the entropy of a black hole in terms of the area of event horizon and Hawking conjectured that black holes emit thermal radiations having a black body spectrum . It brings about an identicalness in the thermal behavior of black holes and the conventional laws of thermodynamics. Classical GR fails to explain it well because in the traditional thermodynamics, entropy is described under the notion of discrete states of a quantum system. It makes black hole entropy phenomenologically important in the search of quantized gravity.

We will also discuss (in causal set cosmology), the problems of fine-tuning and naturalness, which are associated with cosmological constant and can only be addressed faithfully under a theory which accompanies a discrete version of spacetime such as the causal set theory.

What is causal set theory?

In this episode, I will briefly discuss, what causal sets are, without building up the kinematical part? We will have a detailed outlook on the kinematics of causal set theory in future blogs.  

Causal set (causet) program was initiated in early eighties by Rafael Sorkin as a first and direct acknowledgement to the metric recovery theorems[1] proposed by Hawking and Malament in the mid seventies. The idea of causet theory solely relies on two aesthetic concepts of physics: discreteness and causality. Informally, discreteness of spacetime means that space (or the three-dimensional line elements) is composed of a finite numbers of points where each point is assumed to be the smallest fundamental spatial point in nature and time is framed as the shortest possible time interval. Causality is the exquisite essence of the theory in such a way so as any event of nature can influence any other event if it is to the future of the previous since spacelike events remain uninfluential to each other[2]. This is the direct consequence of cause–and-effect relationship and an important feature of causet theory since it prohibits time travel[3].

References

Hawking, S. W., King, A. R., & McCarthy, P. J. (1976). A new topology for curved space–time which incorporates the causal, differential, and conformal structures. Journal of Mathematical Physics, 17(2), 174–181. https://doi.org/10.1063/1.522874
Malament, D. B. (1977). The class of continuous timelike curves determines the topology of spacetime. Journal of Mathematical Physics, 18(7), 1399–1404. https://doi.org/10.1063/1.523436
Bekenstein, J. D. (1973). Black Holes and Entropy. Physical Review D, 7(8), 2333–2346. https://doi.org/10.1103/PhysRevD.7.2333
Hawking, S. W. (1975). Particle creation by black holes. Communications In Mathematical Physics, 43(3), 199–220. https://doi.org/10.1007/BF02345020

  1. We will see in the upcoming posts that how important these theorems are, to the continuum approximation of discrete spacetime.
  2. They are thought to be the part of different universes but it is quite early to say anything until the causet dynamics develop completely.
  3. Causal set theory does not allow closed timelike curves.
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