Physicists do not agree on a maximum temperature, but the Planck temperature of 1.41679 x 10^32 Kelvin is the current best guess. The maximum temperature is a function of particle motion, but reaching the speed of light is impossible. One scientist proposed defining it as the energy of the universe put into the lightest possible particle acceleration close to the speed of light. The Planck temperature is reached in the universe under two conditions: 1 Planck time after the Big Bang and in the final moments of a black hole’s life.
There is no agreed upon value, among physicists, for a maximum possible temperature. According to the current best guess of a complete physical theory, it is the Planck temperature, or 1.41679 x 1032 Kelvin. This translates to approximately 2,538 x 1032° Fahrenheit. Because current theories of physics are incomplete, however, it’s possible it could be hotter.
The answer a typical physicist will give to this question will depend on his implicit opinion of the completeness of the current set of physical theories. Temperature is a function of particle motion, so if nothing can move faster than the speed of light, then the maximum can be defined as a gas whose atomic constituents each move at the speed of light. The problem is that reaching the speed of light in this universe is impossible; the speed of light is a quantity that can only be approached asymptotically. The more energy put into a particle, the closer it gets to moving at the speed of light, although it never quite reaches it.
At least one scientist has proposed defining the maximum possible temperature as what someone would get if they took all the energy in the universe and put it into the lightest possible particle acceleration they could find as close to the speed of light as possible. If this is true, then discoveries about elementary particles and the size/density of the universe could be relevant to discovering the correct answer to the question. If the universe is infinite, there may be no formally defined limit.
While infinite temperature may be possible, it may be impossible to observe it, making it irrelevant. According to Einstein’s theory of relativity, an object accelerated to near the speed of light gains an enormous amount of mass. This is why no amount of energy can be sufficient to accelerate any object, even an elementary particle, to the speed of light: at the limit it becomes infinitely massive. If a particle is accelerated to a certain speed close to the speed of light, it gains enough mass to collapse into a black hole, making it impossible for observers to make any statements about its speed.
The Planck temperature is reached in this universe under at least two separate conditions, according to some theories. The former occurred only once, 1 Planck time (10-43 seconds) after the Big Bang. At the time, the universe existed in an almost perfectly ordered state, with entropy close to zero. It may also have been a singularity, a physical object that can only be described by three quantities: mass, angular momentum, and electric charge. The Second Law of Thermodynamics, however, insists that the entropy (disorder) of a closed system must always increase. This means that the early universe had only one direction to go – that of higher entropy – and suffered an almost instantaneous collapse.
The second set of conditions capable of producing the Planck temperature are those that occur in the final moments of a black hole’s life. Black holes slowly evaporate due to quantum tunneling of matter adjacent to the black hole’s surface. This effect is so small that a typical black hole would take 1060 years to radiate its full mass, but smaller black holes, such as those with the mass of a small mountain, might take as little as 1010 years to evaporate. As a black hole loses mass and surface area, it begins to radiate energy more rapidly, thus heating up, and in the final instant of its existence, it radiates energy so rapidly that it momentarily reaches the Planck temperature.
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