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The article refers to a gravitational something tensor, and, I’m wondering whether that is because this is actually dealing with gravitational effects, or whet

by drdeca 4y ago

The article refers to a gravitational something tensor, and, I’m wondering whether that is because this is actually dealing with gravitational effects, or whether it is just called that because it is the same tensor (or basically the same, or analogous in some way) to the stress energy tensor? It also isn’t obvious to me what “mass radius” ought to mean... I guess like, the average distance from the center of the mass, but like... Idk, I guess that makes sense..? But I guess I’m not sure what the “meaning” of that quantity would be? Like, what makes that a relevant quantity for describing the system? Is it in case you are hoping to describe some gravitational effects? Or...? Hmm... well, I guess one could ask the same question about the “charge radius”... but that one to me sounds like it would have clearer uses? Like, if you are describing the EM forces on/from a charge, then the charge being distributed over a region would have different results than if it were at a single point I’d think. Though, “the center of the proton” also doesn’t have one single position either.. but I imagine one could kind of separate those two things, comparing “what if we had a point particle with the charge and mass of a proton (which of course would be in a superposition over a range of positions)” to “what if we had a proton, with charge distributed about the center (and the center distributed over a range of positions in the same way as the hypothetical point particle)”

evanb 4y ago
A good cartoon is that a form factor is the function that describes how an object deforms when exposed to an outside influence with a particular momentum. The form factor is a function of momentum. There are many different kinds of outside influence. They can be scalar (think: just increasing the pressure uniformly), vector (put in an electric field), tensor (zap with a gravitational wave), pseudovector (magnetic field), pseudoscalar (zap with a pion). Of course, you can apply a scalar outside influence and a vector at once. But the scalar, vector, tensor, pseudovector, and pseudoscalar labels denote different representations of the Lorentz group [lorentz]. What's more: the Wigner-Eckhart theorem [wigner] basically says [cheat] that the response can be factored into three pieces: the strength of the external influence, a factor that depends only on the representation of the external influence, and a factor that depends only on the property of the thing you're talking about (a proton, in this instance). So people call it the gravitational form factor because if you exposed the proton to a gravitational wave, it's the thing you need to know about the proton to know how it deforms. Note that because of the factorization you don't actually have to zap the proton with a gravitational wave! You can measure it by zapping the proton with other stuff, as long as you can get that stuff to have the right rotational properties or measure the response to many different perturbations and sum the responses the right way to mock up a tensor operator. The experiment at JLab doesn't use gravitational waves, it uses these latter approaches. Roughly speaking at zero momentum the form factor is the charge of the object you measure if it's just sitting there. So the electric form factor evaluated at zero momentum is the electric charge, the gravitational form factor evaluated at zero momentum is the mass. What are radii? Express the form factor as a function of momentum^2 [possible]. In units that physicists like to work in (where c=1, hbar=1), the units of momentum are 1/length. Expand the form factor as a Taylor series in momentum^2 and you will get form factor(p) = charge + # radius^2 p^2 + ... where # is a known dimensionless number. The above story is a cartoon but can be made more-or-less precise depending on how much quantum field theory you learn. lorentz: https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group https://en.wikipedia.org/wiki/Representation_theory_of_the_L... wigner: https://en.wikipedia.org/wiki/Wigner%E2%80%93Eckart_theorem https://en.wikipedia.org/wiki/Wigner%E2%80%93Eckart_theorem cheat: this is a little bit of a cheat, it's only true to leading order in a taylor series in the strength of the external influence. possible: it's always possible to arrange this, or at least to separate the momentum dependence into a factor dictated by the rotational symmetry properties and another factor dictated by the object, just like in the Wigner-Eckhart theorem.
- Bjartr 4y ago
  This is an awesome writeup, cartoon or not. Thanks for taking the time to explain!
joe__f 4y ago
Quoting from the article (which quotes from the paper): "Note that I am not even attempting to find an analogy for the gluonic gravitational form factors that would help you understand them. They're described in the paper as "the matrix elements of the energy–momentum tensor of the proton" The energy momentum tensor is the same as the stress energy tensor.

by drdeca 4y ago

evanb 4y ago

A good cartoon is that a form factor is the function that describes how an object deforms when exposed to an outside influence with a particular momentum. The form factor is a function of momentum. There are many different kinds of outside influence. They can be scalar (think: just increasing the pressure uniformly), vector (put in an electric field), tensor (zap with a gravitational wave), pseudovector (magnetic field), pseudoscalar (zap with a pion). Of course, you can apply a scalar outside influence and a vector at once. But the scalar, vector, tensor, pseudovector, and pseudoscalar labels denote different representations of the Lorentz group [lorentz]. What's more: the Wigner-Eckhart theorem [wigner] basically says [cheat] that the response can be factored into three pieces: the strength of the external influence, a factor that depends only on the representation of the external influence, and a factor that depends only on the property of the thing you're talking about (a proton, in this instance). So people call it the gravitational form factor because if you exposed the proton to a gravitational wave, it's the thing you need to know about the proton to know how it deforms. Note that because of the factorization you don't actually have to zap the proton with a gravitational wave! You can measure it by zapping the proton with other stuff, as long as you can get that stuff to have the right rotational properties or measure the response to many different perturbations and sum the responses the right way to mock up a tensor operator. The experiment at JLab doesn't use gravitational waves, it uses these latter approaches. Roughly speaking at zero momentum the form factor is the charge of the object you measure if it's just sitting there. So the electric form factor evaluated at zero momentum is the electric charge, the gravitational form factor evaluated at zero momentum is the mass. What are radii? Express the form factor as a function of momentum^2 [possible]. In units that physicists like to work in (where c=1, hbar=1), the units of momentum are 1/length. Expand the form factor as a Taylor series in momentum^2 and you will get form factor(p) = charge + # radius^2 p^2 + ... where # is a known dimensionless number. The above story is a cartoon but can be made more-or-less precise depending on how much quantum field theory you learn. lorentz: https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group https://en.wikipedia.org/wiki/Representation_theory_of_the_L... wigner: https://en.wikipedia.org/wiki/Wigner%E2%80%93Eckart_theorem https://en.wikipedia.org/wiki/Wigner%E2%80%93Eckart_theorem cheat: this is a little bit of a cheat, it's only true to leading order in a taylor series in the strength of the external influence. possible: it's always possible to arrange this, or at least to separate the momentum dependence into a factor dictated by the rotational symmetry properties and another factor dictated by the object, just like in the Wigner-Eckhart theorem.

Bjartr 4y ago
This is an awesome writeup, cartoon or not. Thanks for taking the time to explain!

joe__f 4y ago

Quoting from the article (which quotes from the paper): "Note that I am not even attempting to find an analogy for the gluonic gravitational form factors that would help you understand them. They're described in the paper as "the matrix elements of the energy–momentum tensor of the proton" The energy momentum tensor is the same as the stress energy tensor.