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From my understanding a faster body has more energy because it can make bigger change in other systems. Also from my understanding temperature is a measurement of average speed of the particles of a given body, so I don't understand the actual difference beyond that in the temperature case the average speed doesn't point anywhere and in an moving body it points.

Also sometimes I hear people saying that particles accelerators can make single protons reach X degrees, does it make sense to talk about temperature at this scale?

Qmechanic
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Edson Luiz Brasil Filho
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2 Answers2

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You can also form a Kugelblitz black hole with fast particles whose kinetic energy is much larger than their rest mass (for example Neutrinos) the same way you can form it with photons.

A single particle can't become a black hole due to its speed, but if you let a bunch of particles collide in a spherical symmetric way their kinetic energy will also contribute to the final mass of the black hole (assuming nothing is radiated away in gravitational waves, but in the spherical symmetric case that doesn't happen anyway).

If the particles fall from rest at infinity (with the negative escape velocity) the kinetic and potential energy add to zero and the mass of the final black hole will simply be the sum of the particles' rest masses. If they fall faster than that the final mass will be higher, and if they fall slower lower.

Everything with less than the escape velocity contributes less than its rest mass to the system, and everything with more contributes more. Therefore a bottle with hot gas weighs more than a bottle with cold gas, although the number of atoms is the same:

Wikipedia wrote: "The invariant mass of any system is measured in a frame where it has zero total momentum, such as a bottle of hot gas on a scale. In such a system, the mass which the scale weighs is the invariant mass, and it depends on the total energy of the system. It is thus more than the sum of the rest masses of the molecules, but also includes all the totaled energies in the system as well."

A single particle cannot become a black hole since its invariant mass doesn't increase with speed:

Wikipedia wrote: "An increase in the energy of such a system which is caused by translating the system to an inertial frame which is not the center of momentum frame, causes an increase in energy and momentum without an increase in invariant mass."

but that's just a side note since you asked for the Kugelblitz where all the momentum vectors add to zero. In that case the answer is yes, the kinetic energy of the particles does contribute to the invariant mass of the resulting black hole.

Yukterez
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The key difference is that temperature doesn’t increase the momentum of the system. When temperature is increased the average kinetic energy of the constituent particles increases, but not the total momentum.

In general relativity the source of gravitation is the stress energy tensor. This includes energy density, but it also includes momentum flux. Momentum flux contributes in a way that prevents simply going fast from producing an event horizon.

Dale
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