0

Under $E=mc^2$, 1kg of matter has $9\times 10^{10}$ joules of energy. So, if I had just the light shining from $9\times 10^8 $ 100 Watt light bulbs inside a perfectly reflective box, would that light have the same amount of inertia as the 1kg of matter? This is a follow up to my similar question earlier this week about the gravity of light.

Is the gravity of light equal to the gravity of mass under $E=mc^2$?

SmarthBansal
  • 1,027
foolishmuse
  • 4,551
  • Your first task is probably to give a precise definition of inertia. What does it mean for something to have more or less inertia? – Luke Pritchett Mar 14 '18 at 17:10
  • The standard definition is fine: "a property of matter by which it continues in its existing state of rest or uniform motion in a straight line, unless that state is changed by an external force." Does it take as much force to push my box of light as it does to push a 1kg mass? You can look at my earlier question about light and gravity to see that this is a followup. – foolishmuse Mar 14 '18 at 17:12
  • You should probably define what you mean when you refer to the force it "takes" to push something. – probably_someone Mar 14 '18 at 17:25
  • @probably_someone: the force required to push the box up to a certain speed, eg. 1 m/s as compared to the force required to push a 1kg mass up to 1 m/s – foolishmuse Mar 14 '18 at 17:36
  • You can use any force to get an object up to a certain speed, as long as you apply that force for the right amount of time. – probably_someone Mar 14 '18 at 17:37
  • @probably_someone: yes, but what I am asking is for the comparison between the box of light and the 1kg mass. The issue of how long it takes is not relevant to the question. But we can just say the force it takes to push it up to 1m/s in 1 sec. It is the comparison between the box of light and the 1kg mass that is the issue here. – foolishmuse Mar 14 '18 at 17:40
  • Ok, so really what you want is the force it takes to produce a certain acceleration. In that case, your concept of "inertia" is just mass, or more precisely, rest energy. – probably_someone Mar 14 '18 at 17:52
  • Also, once we're talking about acceleration we have to notice that light cannot be accelerated. It always travels at the speed of light, so using the ratio of force to acceleration doesn't make sense for light. – Luke Pritchett Mar 14 '18 at 18:00
  • Let me clarify: Light travels in a straight line unless a force (such as gravity) acts on it. Because changing the direction of a beam of light changes its momentum, there needs to be a nonzero transfer of momentum to make this happen. In that sense, light has inertia. We know that gravity and inertia act identically. I just want to confirm that is the case for light. Light does have gravity, does it also have the properties of inertia? – foolishmuse Mar 14 '18 at 18:12
  • Light can carry momentum, and it can transfer its momentum to other objects and receive momentum from other objects. My recommendation is to forget about "inertia" as a property of individual objects; it's not precisely defined and physicists today don't talk about it. – Luke Pritchett Mar 14 '18 at 18:26
  • Think about what you're saying: under your definition, something has "inertia" if you have to change its momentum in order to change its momentum. Since your definition is tautological, every object has this property. – probably_someone Mar 14 '18 at 18:27
  • If you like, "inertia" is a law of motion, not a property of individual objects. – Luke Pritchett Mar 14 '18 at 18:27
  • Perhaps I need to better explain what I have in mind: In the theory of general relativity, the equivalence principle deals with the equivalence of gravitational and inertial mass, and to Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body is the same as the pseudo-force experienced by an observer in a non-inertial frame of reference. So I know that my box of light has gravity. Does it also have inertia in the GR sense? – foolishmuse Mar 14 '18 at 18:40
  • To look back at the answer to the same question about light and gravity, perhaps we can say: "stationary inertial source with a mass of E/c2" even if there is not an actual mass that can be measured. Does this seem correct? – foolishmuse Mar 14 '18 at 19:03
  • 1
    "Under $E=mc^2$, 1kg of matter has $9 \times 10^{10}$ joules of energy." You might want to recalculate. Then we come to "So, if I had just the light shining from $9\times 10^8$ 100 Watt light bulbs [...]". This confuses power and energy. Finally, while pop-sci sources tend to describe GR in the terms you are using the actual theory involves the energy-momentum tensor as the source of gravitation, and while someone who understands the theory can rig cases in which the meaning of the math agrees with the meaning of the words, you can't reliably work the trick in reverse. – dmckee --- ex-moderator kitten Mar 14 '18 at 19:22
  • @dmckee: Recalculate? Yes, decimal problem. If you look back at the question on light and gravity you'll see this answer: "However, if the light in question is bouncing around inside of a perfect mirrored box, and the lengths and time scales of interest are large compared to the size of the box and how long it takes light to bounce from one side of the box to the other, then yes, it would work to treat the light more simply as a stationary gravitational source with a mass of E/c2. " So if it can act as a gravitational source, why not an inertial source? – foolishmuse Mar 14 '18 at 21:04
  • Even in special relativity a box of light has more inertia than a empty box. This follows immediately from a proper definition for the mass of a system in terms of the square of the energy-momentum four-vector of the system. And it is discussed in every even remotely complete source on relativity written in modern terms. – dmckee --- ex-moderator kitten Mar 14 '18 at 23:11
  • @dmckee: Thank you. And is that inertial mass equal to E/c^2 ? – foolishmuse Mar 15 '18 at 15:38

1 Answers1

1

First we might want to think how the energy of "free photon gas" changes according to an observer that changes his frame. One laser beam is traveling to the west and other similar one is traveling to the east. That is our "free photon gas". The observer is first at rest relative to the center of mass of the gas, then the observer accelerates to the east.

Well, the energy changes the same way as the energy of material objects. The details of the change are that one beam loses energy and the other beam gains more energy than the other one loses. Relativistic Doppler shift formula can be used to calculate the change.

Now let's consider light in a box. The box is a long one inside of which one laser beam is traveling to the west and another similar one is traveling to the east. The box and the beams have the same length.

In this case the photons that keep moving to the west gain energy and photons that keep moving to the east lose energy, the amount of change is the same as in the "free light" case for those particular photons. BUT according to the accelerating observer the west beam gains photons from the east beam, and the east beam loses photons to the west beam. So we can see that this trapped light gains more energy than the free light does. So trapped light gains more momentum too. So it has more inertia. In boosts things with large inertia change their momentum a lot.

So from all of that we conclude: One joule of trapped light has more inertia than one joule of free light - which has the same amount of inertia as one joule of matter.

stuffu
  • 1,978
  • 11
  • 11