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| Fig. 1: A universe consisting of particle at the bottom of time-dependent potential will not conserve total energy. (Source: A. Madurowicz) |
In 1915 mathematician and physicist Emmy Noether proved one of the most beautiful and powerful theorems of classical mechanics. When using the Lagrangian formulation, one can calculate the trajectory of a system by using an integral over all of space and all of time of the Lagrangian density function, which describes the relationship between the natural coordinates of the system and a scalar value known as the Action, which the universe should set out to minimize (known as the principle of least action). [1] Noether's Theorem showed that for every differentiable symmetry of the action, there is a corresponding conserved quantity (formally, the conserved quantity is that of the canonically conjugate variable.) [2] For example, one could consider translations in space. Since the laws of physics should not change whether or not you consider them here or there, or anywhere else, we expect the laws of physics to obey spatial translation symmetry, and by applying Noether's Theorem directly to this assumption, one can derive the corresponding conserved quantity: linear momentum. Rotational symmetry gives rise to angular momentum conservation. And perhaps most importantly of all, time translation symmetry mandates the conservation of energy. (See example in Fig. 1.) For the entirety of classical mechanics, this result is quite beautiful and makes a profound impact on our sensibility. The pure fact that the laws of physics ought not to change with time, that the rules are set in stone is alone the only required piece of information that can prove that energy is neither created nor destroyed. But when trying to apply this logic to the universe as a whole, the story becomes more complicated
The model of the universe best supported by empirical evidence is known as ΛCDM, referring to a universe with ordinary matter and also two new components, Cold Dark Matter and the mysterious Λ, the cosmological constant, and is based on Einstein's General Theory of Relativity. While General Relativity is fundamentally based on the assumption that energy is locally conserved, (both the Einstein tensor and the stress-energy tensor are divergence free), it is impossible to define the global energy consistently. When one tries to integrate the stress-energy tensor over all space, one will find that there is no coordinate-system-independent way to do so. [3] The intuition behind this is simple - observers in unique reference frames will simply disagree over the final answer when you sum everything up. What is stationary in one frame may be moving in another, and thus one could boost into an arbitrary frame to give any particle arbitrary energy. In the shift from classical mechanics to relativistic mechanics, we had to throw out the idea that there exists absolute space and time, some preferred coordinate system in which the delicate dance of the laws of nature hold true, and as a consequence we must also throw away the idea of an absolute global energy of the universe. There are some special cases when it becomes possible to define the global energy consistently, notably when the metric is asymptotically flat and static, but this is not generally the case, and indeed not the case for ΛCDM. The metric of our universe, the Friedmann-Lemaître-Robertson-Walker metric, is decidedly time-dependent, as it includes a term called the scale factor a(t), which is necessary to describe the expansion of space over time. [4] Observations of distant supernovae provide one line of experimental evidence for this expansion, and confirm the necessity of Λ for a consistent theory. [5]
When examining the history of the universe, it is straightforward to see how global energy conservation breaks down. Highly energetic photons once dominated the global energy budget, but after the epoch of recombination and the decoupling of the radiation from the plasma, those photons began to redshift as space expanded, and nearly thirteen billion years later those once energetically dominant photons have become the cosmic microwave background, or CMB, which is so cold and energetically void that it takes the most sensitive detectors that humankind can fabricate to detect. If one sits in the reference frame which is stationary with respect to the CMB, the energy is simply lost (although some will try to claim that the radiant energy is converted into gravitational energy through the use of the stress-energy-momentum pseudotensor, but this is not widely accepted.) This once dominant radiation, the CMB, is currently 9 × 10-5 times the critical energy density, less than one hundredth of a percent of the total energy of the universe. Λ, on the other hand, is 0.692 times the critical density, or nearly seventy percent, which currently dominates over all other sources of energy, driving the expansion of space in an accelerating fashion, mysteriously filling space with constant energy. [6] As the most abundant form of energy in existence, if one could find a way to convert the expansion of space into useable energy, one would have a nearly unbounded and limitless source of energy available anywhere in the universe.
© Alexander Madurowicz. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, 2nd Ed. (Addison-Wesley, 2011).
[2] T. H. Boyer, "Derivation of Conserved Quantities from Symmetries of the Lagrangian in Field Theory," Am. J. Phys. 34, 475 (1966).
[3] S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Pearson 2003).
[4] B. Ryden, Introduction to Cosmology (Addison-Wesley, 2002).
[5] A. G. Reiss and A. V. Fillepenko et al., "Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant," Astron. J. 116, 1009 (1998).
[6] P. A. R. Ade et al., "Planck 2015 Results XIII. Cosmological Parameters," Astron. Astrophys. 594, A13 (2016).