arXiv subjects
Publications and source records attributed to Jarmo Mäkelä.
We formulate an argument, based on the use of the action-angle variables and the Bohr-Sommerfeld quantization rule, to the effect that if there exists the smallest possible non-zero area, there also exists, for massive particles, the largest possible observable speed, which is a bit less than the speed of light.
Beginning from the standard Arnowitt-Deser-Misner (ADM) formulation of general relativity we construct a tentative model of quantum gravity from the point of view of an observer with constant proper acceleration, just outside of a horizon of spacetime. In addition of producing the standard results of black-hole thermodynamics, our model makes an entirely new prediction that there is a certain upper bound for the energies of massive particles. For protons, for instance, this upper bound is around $1.1\times 10^{21}eV$. The result is interesting, because this energy is roughly of the same order of magnitude as are the highest energies ever measured for protons in cosmic rays.
Inspired by Einstein's Strong Principle of Equivalence we consider the effects of quantum mechanics to the gravity-like phenomena experienced by an observer in a uniformly accelerating motion in flat spacetime. Among other things, our model of quantum gravity, derived from the first principles, predicts the Unruh effect, and a discrete area spectrum for spacelike two-surfaces.
$χ$iplot is an HTML5-based system for interactive exploration of data and machine learning models. A key aspect is interaction, not only for the interactive plots but also between plots. Even though $χ$iplot is not restricted to any single application domain, we have developed and tested it with domain experts in quantum chemistry to study molecular interactions and regression models. $χ$iplot can be run both locally and online in a web browser (keeping the data local). The plots and data can also easily be exported and shared. A modular structure also makes $χ$iplot optimal for developing machine learning and new interaction methods.
Existing methods for explaining black box learning models often focus on building local explanations of model behaviour for a particular data item. It is possible to create global explanations for all data items, but these explanations generally have low fidelity for complex black box models. We propose a new supervised manifold visualisation method, SLISEMAP, that simultaneously finds local explanations for all data items and builds a (typically) two-dimensional global visualisation of the black box model such that data items with similar local explanations are projected nearby. We provide a mathematical derivation of our problem and an open source implementation implemented using the GPU-optimised PyTorch library. We compare SLISEMAP to multiple popular dimensionality reduction methods and find that SLISEMAP is able to utilise labelled data to create embeddings with consistent local white box models. We also compare SLISEMAP to other model-agnostic local explanation methods and show that SLISEMAP provides comparable explanations and that the visualisations can give a broader understanding of black box regression and classification models.
Causal structure discovery (CSD) models are making inroads into several domains, including Earth system sciences. Their widespread adaptation is however hampered by the fact that the resulting models often do not take into account the domain knowledge of the experts and that it is often necessary to modify the resulting models iteratively. We present a workflow that is required to take this knowledge into account and to apply CSD algorithms in Earth system sciences. At the same time, we describe open research questions that still need to be addressed. We present a way to interactively modify the outputs of the CSD algorithms and argue that the user interaction can be modelled as a greedy finding of the local maximum-a-posteriori solution of the likelihood function, which is composed of the likelihood of the causal model and the prior distribution representing the knowledge of the expert user. We use a real-world data set for examples constructed in collaboration with our co-authors, who are the domain area experts. We show that finding maximally usable causal models in the Earth system sciences or other similar domains is a difficult task which contains many interesting open research questions. We argue that taking the domain knowledge into account has a substantial effect on the final causal models discovered.
We obtain a simple relationship between the change in the Brown-York energy inside of a closed two-surface just outside of a horizon of spacetime, and the change in the area of that two-surface.
As an attempt to realize Wheeler's "it-from-bit proposal" that physics should be reduced to simple yes-no questions we consider a model of loop quantum gravity, where the only allowed values of the quantum numbers $j_p$ at the punctures of the spin network on the spacelike two-surfaces of spacetime are $0$ and $\frac{1}{2}$. When $j_p=0$, the puncture is in the vacuum, and it does not contribute to the area of the two-surface, whereas when $j_p=\frac{1}{2}$, the puncture is in an excited state, and the allowed values of the associated quantum number $m_p$ are $-\frac{1}{2}$ and $+\frac{1}{2}$. As a consequence, the spin network used as a model of spacetime is analogous to a system of particles with spin $\frac{1}{2}$, and every puncture carries exactly one bit of information. When applied to spacetimes with horizon our model enables us to find an explicit expression for the partition function of spacetime. Using this partition function we may, among other things, obtain the Bekenstein-Hawking entropy law for black holes. When applied to cosmological models with horizon the partition function predicts a cosmic phase transition in the early Universe, where the cosmological constant went through a dramatic decrease and the matter of the Universe was created out of the vacuum.
We point out that with a specific counting of states loop quantum gravity implies that black holes perform a phase transition at a certain characteristic temperature $T_C$. In this phase transition the punctures of the spin network on the stretched horizon of the black hole jump, in effect, from the vacuum to the excited states. The characteristic temperature $T_C$ may be regarded as the lowest possible temperature of the hole. From the point of view of a distant observer at rest with respect to the hole the characteristic temperature $T_C$ corresponds to the Hawking temperature of the hole.
The choice of the area operator in loop quantum gravity is by no means unique. In addition to the area operator commonly used in loop quantum gravity there is also an area operator introduced by Krasnov in 1998, which gives uniformly spaced area spectra for the horizons of spacetime. Using Krasnov's area operator we consider the thermodynamics of spherically symmetric spacetimes equipped with horizons in loop quantum gravity. Among other things, our approach implies, in a pretty simple manner, that every horizon of spacetime emits thermal radiation and possesses emtropy which, in the natural units, is one-quarter of its area. When applied to the de Sitter spacetime loop quantum gravity provides an explanation both to the presence and the smallness of the cosmological constant.
We consider a novel model of cosmic inflation. In our model one does not need any specific matter field to drive inflation, but inflation stems from the microscopic, Planck scale structure of spacetime, thus being of quantum gravitational origin. At a certain temperature spacetime performs a phase transition, where the cosmological constant drops from a huge, Planck scale value, which is about $10^{87}s^{-2}$ to its present, pretty small value $10^{-35}s^{-2}$. When the cosmological constant is large, the universe goes through a period of very rapid expansion which, however, comes to an abrupt end after the phase transition has been completed. Assuming that the cosmological constant depends on the age of the universe in an appropriate manner during the phase transition one may recover the predictions of the conventional inflationary scenario.
We consider a possibility that the formally infinite vacuum energy of the quantized matter fields could be stored into Planck-size quantum black holes acting as the fundamental constituents of space and time. Using the recently proposed thermodynamical equation of state obeyed by the cosmological constant we indicate, how this idea might explain the smallness of the cosmological constant, if the vacuum energy is the source of the cosmological constant.
We consider a possibility that the cosmological constant may not be a constant, but a free thermodynamical variable. To this end we construct a microscopic model of a spacelike two-sphere just inside of the cosmological horizon of the de Sitter spacetime. In our model the spacelike two-sphere in question is assumed to consist of discrete constituents, each of them contributing to the two-sphere an area, which is in an integer times a constant. Using our model we obtain a thermodynamical equation of state for the cosmological constant. Our equation of state implies that the cosmological constant has a certain positive lower bound, which depends on the number of the constituents of the two-sphere.
We consider a microscopic model of a stretched horizon of the Schwarzschild black hole. In our model the stretched horizon consists of a finite number of discrete constituents. Assuming that the quantum states of the Schwarzschild black hole are encoded in the quantum states of the constituents of its stretched horizon in a certain manner we obtain an explicit, analytic expression for the partition function of the hole. Our partition function predicts, among other things, the Hawking effect, and provides it with a microscopic, statistical interpretation.
We seek for an alternative to the metric tensor $g_{μν}$ as a fundamental geometrical object in four-dimensional Riemannian manifolds. We suggest that the metric tensor $g_{μν}(P)$ at a given point $P$ of a manifold may be replaced by a four-dimensional geometrical simplex $σ^^4(P)$ embedded to the tangent space $T_P$ of the point $P$. The number of two-faces, or triangles, of $σ^4(P)$ is the same as is the number of independent components of $g_{μν}(P)$, and hence we may replace the components of $g_{μν}$ by the two-face areas of $σ^4(P)$. In this sense the concept of distance may, in four-dimensional Riemannian manifolds, be reduced to the concept of area. This result may find some applications in the thermodynamical approaches to quantum gravity.
We point out that certain equations which, in a very recent paper written by E. Verlinde, are postulated as a starting point for a thermodynamical derivation of classical gravity, are actually consequences of a specific microscopic model of spacetime, which has been published earlier.
We note that if we construct from the observed anomalous acceleration $a_P=(8.74\pm1.33)\times 10^{-10}m/s^2$ of the spacecrafts Pioneer 10 and 11 towards the Sun, the proton mass $m_p$ and Newton's gravitational constant $G$ a quantity $\ell_P:=(Gm_p/a_P)^{1/2}$, which has the same dimension as length, then $\ell_P$ is roughly of the same order of magnitude as is the Compton wave length of a proton. We formulate a simple quantum mechanical hypothesis, which aims to provide an explanation to this interesting numerical coincidence.