Is there any experimental evidence that pure energy (massless) can curve spacetime?
There is no such thing as "pure energy". Energy is a property we use to describe how the world behaves around us. You could just as well ask for "pure charge" or "pure angular momentum".
Let me rephrase: is there experimental evidence that a particle with 0 rest-mass by itself will warp spacetime?
This depends on what you mean. Gravitational influence on the elementary particle level has not been observed for massive nor massless particles. As a collective phenomenon, the early Universe would have behaved very differently if radiation did not have a gravitational influence.
My quantum theory is too weak for this to be more than a question: is nuclear binding energy an example? Gluons are massless, I think, yet the mass of a proton is considerably greater than three quarks.
It has been shown that neutrons fall (McReynolds; Collela, Overhauser and Werner) , and that photons are redshifted (Pound-Rebka).
This is the effect of space-time curvature on elementary particles, the OP wanted evidence of the effect of "pure energy" on space-time.
Do you know of any experimental evidence that a particle with non-zero rest mass will warp space-time?
Yes. It's a good example, and you beat me to it.
So - if only massive particles produced spacetime curvature then "gravitational charge" would depend on nucleon number (or very very nearly so) not mass. Thus the Eotvos experiments would not have the outcome they did.
Well, charge is frame invariant, unlike energy/momentum.
There are also really perverse consequences of assuming that massless particles are not a source of gravity. Consider that you had an electron + positron plasma contained in a reflective shell. You would predict that the overall body would loose the weight of the electrons and positrons as they annihilate. Thus, a closed system would lose externally measurable mass.
Well, as pointed out, there is no such thing as pure energy. But you said "Gravitational influence on the elementary particle level has not been observed for massive nor massless particles." It unquestionably has for neutrons - massive particles. They fall. the photon story is more complicated, but it is still evident that single photons feel a gravitational field.
If you are talking about producing a gravitational field, conservation of momentum requires this. If your question is "can a single neutron cause a measurable gravitational field", the answer is "measurable? no". Of course the same is true for a grain of rice. Nobody doubts that rice gravitates.
I do not doubt that light gravitates. As mentioned already, if it did not it would have perverse consequences. I am just saying it has not been explicitly tested at the elementary particle level.
Well, in order to have a good experimental test (say a cosmological test), we'd need a theory in which energy does not "curve space-time". I will interpret "curve space-time" here as asking about a theory of gravity. This theory would not be General Relativity, in General Relativity energy is part of the stress-energy tensor, and the theory predicts that energy (such as light) does curve space-time.
The most straightforwards theory where energy does not curve space-time would probably be Nordstrom's theory, https://en.wikipedia.org/wiki/Nordström's_theory_of_gravitation
This theory is definitely incompatible with experiment, it predicts no deflection of light via gravity, and we've observed that gravity does deflect light. So there is experimental evidence that Nordstrom's theory is wrong.
So, are there other theories of gravity in which energy (say light) does not curve space-time? It's hard to say. It's rather problematical to imagine a theory where gravity deflects light (which is observed), and light has momentum (which is also observed - does this point need any emphasis?), and in which the change in momentum of the light does not imply a change in momentum of the gravitating source that deflects the light. It appears to me that one would need a theory that doesn't conserve momentum that's compatible with experiment - I'm not aware of any such theory. There is a certain lack of rigor in the argument, but it doesn't look promising.
Momentum conservation in general relativity is a bit tricky, but in an asymptotically flat space-time (representing isolated system) there are applicable notions of momentum which are conserved when light is deflected by a massive body.
In any event, before we can talk about experimental tests, we need a consistent theory to test.
I think we are using the term "gravitate" differently. I am using it to mean "is affected by gravity". Are you using it to mean "is a source of gravity"?
Since the thread OP was asking about whether massless energy can curve spacetime, I think the assumption is that "source of gravity" is what is meant.
That was the first time I used the word gravitate. Since the OP is directly asking about the influence of massless particles on the curvature of space time, I do not see how my posts can be interpreted as being about the influence of the space time geometry on the particles.
To make things clear: I do not doubt either of the effects. The effects of individual elementary particles on the curvature of space time has not been tested directly - although we have every reason to believe it is there.
I think detecting the gravitational effect of a single particle ( massless or not ) is monumentally difficult, since the effect is so small. It also makes me wonder just what - given quantum effects such as position/momentum uncertainty, the superposition principle etc - that effect would really look like.
However, it should at least in principle be possible to observe the gravitational effects of massless entities such as electromagnetic fields. I am getting a little ahead of current technology here, but suppose we have a in-spiralling binary of a neutron star with some companion, and we examine the gravitational wave signature of that event - the presence of the neutron star's magnetic field ( i.e. its gravitational influence ) should have a detectable effect on the wave form, right ?
This is just a thought experiment, I am well aware of the technical difficulties.
Dark energy seems to be doing a good job of expanding the universe and everything in it. Does that not count?
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