# Nonconservation of Energy and Loss of Determinism

I. Infinitely Many Balls David Atkinson, Porter Johnson
An infinite number of elastically colliding balls is considered in a classical, and then in a relativistic setting. Energy and momentum are not necessarily conserved globally, even though each collision does separately conserve them. This result holds in particular when the total mass of all the balls is finite, and even when the spatial extent and temporal duration of the process are also finite. Further, the process is shown to be indeterministic: there is an arbitrary parameter in the general solution that corresponds to the injection of an arbitrary amount of energy (classically), or energy-momentum (relativistically), into the system at the point of accumulation of the locations of the balls. Specific examples are given that illustrate these counter-intuitive results, including one in which all the balls move with the same velocity after every collision has taken place.

II: Colliding with an Open Set
An actual infinity of colliding balls can be in a configuration in which the laws of mechanics lead to logical inconsistency. It is argued that one should therefore limit the domain of these laws to a finite, or only a potentially infinite number of elements. With this restriction indeterminism, energy non-conservation and (creatio ex nihilo) no longer occur. A numerical analysis of finite systems of colliding balls is given, and the asymptotic behavior that corresponds to the potentially infinite system is inferred.

# Exploring Gravitational Lensing Einstein’s derivation of the lensing equation, solution, and amplification in AEA 62-275 (Albert Einstein Archives, The Hebrew University of Jerusalem, Israel)

Tilman Sauer, Tobias Schütz
In this article, we discuss the idea of gravitational lensing, from a systematic, historical and didactic point of view. We show how the basic lensing equation together with the concepts of geometrical optics opens a space of implications that can be explored along different dimensions. We argue that Einstein explored the idea along different pathways in this space of implication, and that these explorations are documented by different calculational manuscripts. The conceptualization of the idea of gravitational lensing as a space of exploration also shows the feasibility of discussing the idea in the classroom using some of Einstein’s manuscripts.

# Black Hole Entropy is Thermodynamic Entropy Schematic illustration of a black hole Carnot cycle. The system consists of a black hole and a photon gas, enclosed in a box. The size of the black hole is proportional to the temperature of the system, i.e. small is hot and large is cold.

Carina E. A. Prunkl, Christopher G. Timpson
The comparison of geometrical properties of black holes with classical thermodynamic variables reveals surprising parallels between the laws of black hole mechanics and the laws of thermodynamics. Since Hawking’s discovery that black holes when coupled to quantum matter fields emit radiation at a temperature proportional to their surface gravity, the idea that black holes are genuine thermodynamic objects with a well-defined thermodynamic entropy has become more and more popular. Surprisingly, arguments that justify this assumption are both sparse and rarely convincing. Most of them rely on an information-theoretic interpretation of entropy, which in itself is a highly debated topic in the philosophy of physics. We discuss some of the pertinent arguments that aim at establishing the identity of black hole surface area (times a constant) and thermodynamic entropy and show why these arguments are not satisfactory. We then present a simple model of a Black Hole Carnot cycle to establish that black hole entropy is genuine thermodynamic entropy which does not require an information-theoretic interpretation.

# Lost in Math?

Lost in Math? A review of ‘Lost in Math: How Beauty Leads Physics Astray’, by Sabine Hossenfelder

Jeremy Butterfield
This is a review of Hossenfelder’s book, ‘Lost in Math: How Beauty Leads Physics Astray’. The book gives a breezy exposition of the present situation in fundamental physics, and raises important questions: both about the content of the physics, and the way physics research is organized. I first state my main disagreements. Then, I mostly praise the book: I concentrate on Hossenfelder’s discussion of supersymmetry, naturalness and the multiverse.

# After primordial inflation

D. V. Nanopoulos, K. A. Olive, M. Srednicki
We consider the history of the early universe in the locally supersymmetric model we have previously discussed. We pay particular attention to the requirement of converting the quanta of the field which drives primordial inflation (inflatons) to ordinary particles which can produce the cosmological baryon asymmetry without producing too many gravitinos. An inflaton mass of about 1013 GeV (a natural value in our model) produces a completely acceptable scenario.

# An introduction to the classical three-body problem Lagrange’s periodic solution with three bodies at vertices of equilateral triangles. The constant ratios of separations
are functions of the mass ratios alone

Govind S. Krishnaswami, Himalaya Senapati
The classical three-body problem arose in an attempt to understand the effect of the Sun on the Moon’s Keplerian orbit around the Earth. It has attracted the attention of some of the best physicists and mathematicians and led to the discovery of chaos. We survey the three-body problem in its historical context and use it to introduce several ideas and techniques that have been developed to understand classical mechanical systems.