# Conservation of momentum if kinetic energy is converted to mass

There is a moving object. Through an unspecified (science fiction) mechanism its kinetic energy is converted to mass and the object comes to rest.

The mechanism is fictional but in good scifi it is good to adhere to the laws of nature. Does the conversion of kinetic energy to mass violate conservation of momentum? Or is conservation of momentum just a case of conservation of energy, which is conserved when converted to mass?

• Where is your KE converter mounted? Is it on a planet, or a spaceship? How heavy is the target you're trying to stop? If it's relatively light compared to your ship, then you may be able to have your ship absorb the momentum. Jan 23 '19 at 0:42
• I want to decrease velocity (possibly not all the way to 0) by turning energy into mass. If momentum is p and p = mv, can't I keep p the same by increasing m and decreasing v. Jan 23 '19 at 16:24
• No, that doesn't work, unless I made a dumb mistake with the algebra. ;) That equation for momentum is from Newtonian mechanics, but since we're converting energy to mass we should use the relativistic equation, $p=mv\gamma$, where $\gamma=1/\sqrt{1-\frac{v^2}{c^2}}$. To calculate energy we can use $E^2=(pc)^2+(mc^2)^2$, or equivalently, $E=mc^2\gamma$. See en.wikipedia.org/wiki/Energy%E2%80%93momentum_relation and en.wikipedia.org/wiki/Four-momentum Jan 23 '19 at 20:45
• Why not convert the KE to light? That makes it easy to conserve energy & momentum. FWIW, in this answer, I show essentially the reverse process, converting mass to KE & light. Jan 23 '19 at 20:55

Momentum was non-zero before the conversion and zero after the conversion and so conservation of momentum was violated. Therefore, your described conversion is not consistent with the known laws of physics which require conservation of momentum to always hold.

Through an unspecified (science fiction) mechanism its kinetic energy is converted to mass and the object comes to rest.

That part is fine. Einstein gives us the equivalence between mass and energy. So converting the kinetic energy to mass is quite doable. Any form of energy storage will do this.

Does the conversion of kinetic energy to mass violate conservation of momentum?

No. Energy and momentum are different. Momentum must be conserved as well. The total mass/energy of the system before stopping and after stopping must have the same momentum.

Imagine two astronauts floating towards each other in space, one with velocity $$v$$, the other with $$-v$$ and both with the same mass $$m$$. In this reference frame, the total momentum is $$mv + m(-v) = 0$$ and the total kinetic energy is $$mv^2/2 + m(-v)^2/2 = mv^2$$ (in classical mechanics, ignoring special relativity for now). If the astronauts now collide (thud!) and grab each other, they will be at rest afterwards. This is because of conservation of momentum: If $$u$$ is the velocity of both astronauts moving together after the collision, we get $$mv + m(-v) = 0 = (m + m)u \implies u = 0$$. But now the kinetic energy is $$(m + m)u^2/2 = 0$$. Because of conservation of energy, the original kinetic energy $$mv^2$$ must have gone somewhere; it must have been transformed into internal energy in the two astronauts. In this case, most likely into heat in the astronauts' bodies and space suits, but in the case of subatomic particles, it would have been mass (actually, the difference is not clear-cut; as far as relativity is concerned, any internal energy is mass, whether it be heat, internal motion, molecular or atomic binding energy, intrinsic mass coming from the Higgs field...).

So there you are: The kinetic energy of the first astronaut can be converted into internal energy (mass), but only by colliding with something of equal and opposite momentum.

If you're looking for sci-fi writing prompts, the first astronaut could also slow down by ejecting something to carry away all of their momentum, e.g. rocket fuel or a huge blast of radiation. However, this can never result in the astronaut gaining mass, as I'll show below (this time with special relativity, so feel free to skip):

Imagine the astronaut traveling at speed $$v$$, thus having momentum $$p = \gamma mv$$ and total (mass + kinetic) energy $$E = \gamma mc^2$$, where $$\gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}$$. It now ejects something, let us call it radiation, with mass $$M$$, carrying away all its momentum $$p$$, so that it comes to a halt (by conservation of momentum). The energy of the radiation is then $$\Delta E = \sqrt{(Mc^2)^2 + (pc)^2} \geq pc,$$ and by conservation of energy, the astronaut's total energy afterwards must be $$E - \Delta E$$. The mass afterwards is just this divided by $$c^2$$, since the astronaut's momentum is zero: $$m' = \gamma m - \frac{\Delta E}{c^2} \leq \gamma m - \frac{p}{c} = \gamma m \left(1 - \frac{v}{c}\right).$$ Abbreviating $$\beta = \frac{v}{c}$$ and noting that $$\beta < 1$$ (the astronaut is traveling slower than light), we get $$\frac{m'}{m} \leq \gamma(1 - \beta) = \frac{1 - \beta}{\sqrt{1 - \beta^2}} = \sqrt{\frac{1 - \beta}{1 + \beta}} < 1,$$ showing that the astronaut always loses mass.

Since the astronaut (or maybe we should call it a spaceship now) has to collide with something in order to convert its kinetic energy into mass, we might speculate on what... The cosmic microwave background consists of photons coming from all directions in space. So technically, we are all constantly colliding with CMB photons and slowing down relative to the CMB rest frame, but the effect is minuscule. But what if your spaceship had some highly advanced equipment to be able to interact with other things in the environment? Things like dark matter (effect would probably also be minuscule), particles from parallel universes, whatever makes up the vacuum energy of empty space... This is where established physics ends and your creativity starts!

Through an unspecified (science fiction) mechanism its kinetic energy is converted to mass and the object comes to rest.

Does the conversion of kinetic energy to mass violate conservation of momentum?

Yes, in the example you gave.

Kinetic energy can, in theory, be converted to mass, but only if momentum is conserved.

In nature, it is much more common for mass to be converted into kinetic energy. For example, in radioactive decay, some of the mass of a nucleus at rest is converted into kinetic energy of the particles after the decay. But, crucially, there have to be multiple particles afterwards, and their momenta have to cancel each other (e.g. for two particles, they have to go in opposite directions).

Momentum and energy are always separately conserved.

• I am now certain that I gravely misunderstand momentum. Can you lay out an example where the kinetic energy of the moving object is converted to mass, but momentum is conserved? Jan 22 '19 at 13:07
• @Willk The reaction of protons to create an anti-proton: $p+p\to p+p+p+\bar{p}$ With enough KE in the system, two new massive particles are created, but the momentum (and total energy including mc$^2$) of the system will be constant. Jan 22 '19 at 22:48
• @Willk, imagine a flying bullet get stopped by a thick wooden desk at rest, suspended on a thread from above. The kinetic energy gets converted to rest energy of the resulting desk with bullet inside. Momentum remains the same, the desk moves in the original direction, just slower. Jan 23 '19 at 0:01

I'm afraid the idea that you have in mind is not salvagable.

To explain let me refer to the case that is is discussed in the answer by Elias Riedel Gårding.

When the direction is: conversion of energy to kinetic energy then multiple particles have to moving away from each other, with the respective momenta summing to the same value as before the conversion.

This illustrates that any process that proceeds in the reverse direction must be in the form of one or more head on collisions.

So let's say you have a story where a large object slows down by way of a series of head on encounters, in the process harvesting all the kinetic energy. Actually, nowadays we have quite a lot of technology that does just that: harvesting kinetic energy so that it can be reused. In today's electric cars it is pretty much standard to have regenerative braking. When the driver eases off the accelerator pedal the electric motor starts acting as electric generator. Thus energy is harvested, and put back in the car's battery. In the story the incoming objects are coming in on a head on collision trajectory, but instead of colliding the incoming objects are slowed down in an energy harvesting way. That takes care of the momentum conservation issue.

Next, the question of storing the harvested energy. In the story idea the harvested energy is stored in the form of mass. That is: you raise the question: are the laws of physics such that in principle it is possible to make use of some form of nuclear physics to achieve energy storage?

I think that is inherently impossible. I think that with anything you might try the laws of physics are such that you lose more than you gain.
Example:
As we know, in our Sun there are nuclear fusion processes going on. The net result is one Helium nucleus for four Hydrogen nuclei. As we know, in this process mass is converted to energy; the energy source of stars is conversion of mass to energy. Now let's raise the question: do the laws of physics allow a reverse process of that so that you could start with Helium nuclei, pump in energy, and harvest Hydrogen nuclei? My answer: I don't think so. The normal flow of this process is what happens in the Sun, fusion resulting in Helium nuclei. To reverse the direction of flow here is in principle possible - but the cost is such that you lose more than you gain. The reverse process is not impossible, rather it is highly improbable. If you could continuously withdraw Hydrogen nuclei you could make the rate of the reverse process exceed the rate of the normal process. But the act of selecting and withdrawing the Hydrogen nuclei is itself something that requires energy expenditure. (The above description is so crude that I probably confused more than explained.) Anyway, the gist of my story is that I think that no matter what you try you lose more than you gain.

No doubt there are other mechanisms that can be proposed, but my best guess is that each of those will suffer the same problem. It attempts to reverse the direction of a process that is very hard to reverse. That doesn't make it impossible, but making it happen will require so much expenditure of energy that it becomes pointless.

• Fusion of elements iron and heavier is endothermic - this fusion turns the heat energy of supernovae to mass. But I am skeptical about the statement "that takes care of momentum conservation". I suspect your misunderstanding of momentum is much like mine. Jan 22 '19 at 23:33
• In the car example, the way the energy is harvested doesn't make any difference for momentum conservation. Ultimately, all the car's momentum has to be transferred to something else, for example motion of air molecules hitting the windshield or, more importantly, a very slight change in the motion of Earth itself through space! Regenerative breaking just means covering some of the kinetic energy into chemical energy in the battery instead of heat. Momentum conservation is very powerful because while energy comes in many different forms, momentum has only one form: motion. Jan 23 '19 at 23:56