I Testing GR to What Energy Level?

[Arcturus7]

When we talk about energy scales, we often hear that we need new theories of gravity and particle physics when we hit the barrier that is the Planck scale. In particle physics we've only probed up to about 13TeV, which is way WAY below that level. It's easy for us to designate the energy to which we've probed, since we built the LHC and we know what energy it is operating at.

My question is, what sort of energy level have we tested GR to? I'm guessing that the highest energy scales belong to the binary BH mergers detected by LIGO, seeing as the energy emitted from those mergers was colossal, but how do we pin down an energy scale in the same way as with particle physics? Where can we say with confidence that "GR is definitely valid up until xyz energy."?

 

[PeterDonis]

what sort of energy level have we tested GR to?

First we have to be clear about what "energy level" means in this context. When particle physicists talk about energy scales, they really mean energy per particle. That's easy to measure in a device like the LHC, as you say; but it doesn't tell us how to compare that number with, say, some equivalent parameter in a binary BH merger.

The usual way of expressing "scale" in GR is the scale of spacetime curvature, or equivalently of energy density (the equivalence comes from the Einstein Field Equation). For a black hole merger where the total mass of the holes is approximately $M$, the spacetime curvature at the horizon (of the merged hole), which is the relevant parameter for determining, roughly, how strong the GWs are that are emitted, is of order $1 / M^2$. Notice that this is inversely proportional to $M$--i.e., larger holes have smaller spacetime curvature at their horizons.

If you run the numbers for a black hole of 50 solar masses, roughly the size of the merged holes from the LIGO detections (we're only making order of magnitude calculations here so the exact numbers aren't important), the spacetime curvature at the horizon corresponds to an energy density of about $2 \times 10^{34}$ Joules per cubic meter. This is equivalent to a mass density of about $10^{14}$ grams per cubic centimeter, which is roughly the same as the density in neutron stars. So that is a reasonable estimate of the largest curvatures/densities at which GR has been confirmed.

In order to equate these densities with something from particle physics, you would have to convert the energy per particle numbers from something like the LHC into energy densities. Off the top of my head I'm not sure how you would do that, but it might be possible to find estimates someone has made.

 

[Arcturus7]

@PeterDonis I suspected the answer wouldn’t be cut and dried - things never are. Thanks for your helpful answer though!

I was having the same trouble trying to work out a way to put them on an even keel. I basically was trying to find out some sort of quantitative way to compare our “confidence” in the two theories, if you like. My rationale being: we’re still miles and miles away from testing the Standard Model at energies even close to the PE, but I don’t know how correspondingly far away we are with general relativity.

I’ve got to write a mock research proposal for a uni assignment and I’m wondering whether we could use GR and early universe Cosmology to help potentially constrain BSM Physics in some way, so I wanted to know which we had tested to higher energies.

Could both be converted to a temperature scale? I’m fairly sure the accelerator energies can, but the BH merger might be a bit shakier.

 

[PeterDonis]

we’re still miles and miles away from testing the Standard Model at energies even close to the PE, but I don’t know how correspondingly far away we are with general relativity.

As a rough statement, I would say we're at least as far from testing GR at Planck scale energies as we are from testing the Standard Model at such energies; possibly much further.

The obvious Planck scale for gravity is the Planck energy density, which is one Planck energy (Planck mass times $c^2$) per Planck volume. That corresponds to an energy density of $5 \times 10^{113}$ Joules per cubic meter, or $5 \times 10^{93}$ grams per cubic centimeter. That's 79 orders of magnitude larger than the densities I calculated in my previous post.

By comparison, the Planck energy is about $10^{28}$ eV, and the LHC scale is about $10^{13}$ eV, a difference of only 15 orders of magnitude. Even if we argue that, to convert this to an energy density difference, we must raise it to the fourth power (heuristically, this is because in QFT, energy corresponds to inverse length, so energy density corresponds to energy per length cubed, or energy times energy cubed, or energy to the fourth power), this still means a difference of only 60 orders of magnitude, vs. 79 above.

 

[Arcturus7]

As a rough statement, I would say we're at least as far from testing GR at Planck scale energies as we are from testing the Standard Model at such energies; possibly much further.

The obvious Planck scale for gravity is the Planck energy density, which is one Planck energy (Planck mass times $c^2$) per Planck volume. That corresponds to an energy density of $5 \times 10^{113}$ Joules per cubic meter, or $5 \times 10^{93}$ grams per cubic centimeter. That's 79 orders of magnitude larger than the densities I calculated in my previous post.

By comparison, the Planck energy is about $10^{28}$ eV, and the LHC scale is about $10^{13}$ eV, a difference of only 15 orders of magnitude. Even if we argue that, to convert this to an energy density difference, we must raise it to the fourth power (heuristically, this is because in QFT, energy corresponds to inverse length, so energy density corresponds to energy per length cubed, or energy times energy cubed, or energy to the fourth power), this still means a difference of only 60 orders of magnitude, vs. 79 above.

Oh wow - okay thanks! I’m very surprised to see that we’re so far away. I would have expected both to be fairly similar, or even for GR to be further ahead purely because of the difficulty of observing weak gravitational signals. Very interesting stuff.

 

[mfb]

Another way to compare these numbers is via the temperature of the black holes, or their radius compared to the Planck mass. The result is similar - the temperature of stellar mass black holes is tiny even compared to room temperature (and even more compared to LHC temperatures), the radius is very large compared to the length scales of high energy physics and far away from the Planck scale.