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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, would that light have the same amount of gravity as the 1kg of matter?

Qmechanic
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foolishmuse
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  • I don't think light itself has gravity. Since light consists of photons and photons are massless, light does not have mass. Hence, light does not produce gravity. – ConfusedStudent Mar 12 '18 at 19:02
  • If you go to this link it says that energy does have gravity: https://van.physics.illinois.edu/qa/listing.php?id=28195 In particular it says General Relativity says that any form of energy is a source of gravity. Imagine a bag of light, traveling all directions so it has no net momentum. From the outside, that bag has energy but no momentum, exactly like a particle with rest mass that's not moving (in our frame). Its gravitational field looks just like that of a rest mass. I guess I should adapt the question to include light and heat from my light bulbs. – foolishmuse Mar 12 '18 at 19:24

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No, a photon with energy $E$ behaves differently with respect to gravity than a slow-moving object of mass $m=E/c^2$ does.

In fact, that difference was behind one of the first tests of general relativity. General relativity is very nearly consistent with Newtonian gravity when it comes to slow moving objects in our solar system. However, if you use Newtonian gravity, and treat a photon passing by the sun as being an object of mass $E/c^2$, the amount by which the photon should be deflected is off by a factor of $2$ from what general relativity predicts. During the solar eclipse of 1919, Arthur Eddington made measurements of the deflection of starlight which passed close to the sun. His results were consistent with general relativity, and inconsistent with what was predicted with the combination of Newtonian gravity and $E=mc^2$.

Light also behaves differently from a slow-moving mass as far as being a source of gravity. General relativity describes gravity as involving the curvature of spacetime. The source of that curvature is the stress–energy tensor, which has 16 components, instead of being just one number (mass) that's the source of gravity according to Newtonian gravity. A photon's energy and momentum contribute differently to the stress-energy tensor than does an object with a non-zero rest mass.

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 $E/c^2$.

Red Act
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  • So are you saying that the light will have about half of the gravity of the 1kg mass? With the reason being the other 15 components of the stress-energy tensor? – foolishmuse Mar 12 '18 at 20:40
  • This is true but not necessarily complete. If you have a mirrored box at rest with some amount of light bouncing around inside, that light contributes to the box's mass exactly as $E/c^2$. – Rococo Mar 12 '18 at 21:07
  • @foolishmuse A beam of light is deflected by the sun by twice as much as is predicted by Newtonian gravity and $E=mc^2$, but it'd be an oversimplification to say that "light has twice the gravity of its equivalent mass". "How much gravity an object has" is in general more complicated than what can be expressed by a single number. And when you're looking at a particle's motion near a gravitating body, gravity doesn't even behave like a force, i.e., it doesn't behave as simply as $F=ma$, even after taking SR into account. General relativity and Newtonian gravity are completely different beasts. – Red Act Mar 12 '18 at 21:35
  • @Rococo I'm of course aware of that. I suppose that might have been useful to mention, but the question doesn't ask about light bouncing around in a mirrored box, so I didn't address that situation. – Red Act Mar 12 '18 at 21:40
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    @Rococo Eh, I changed my mind. What you pointed out would be a useful part of the answer, so I'll add it to my answer. – Red Act Mar 12 '18 at 21:43
  • @RedAct: Thanks for that. Now we need to know, is the inertia of our box of light the same as the inertia of my original 1kg mass? – foolishmuse Mar 13 '18 at 15:14