Every good nerd knows that E=mc^2.

Every great nerd knows that, really, E^2=m2c4+p2c2.

But what does that even mean?

Oh, I'll tell you, but today I'd like to invite you to an even higher level of nerdom.

There may be even more add ons to Einstein's famous equation that will make even the greatest nerds quiver in their ... space time merch if they turn out to be real.

Because they could point the way to the quantum structure of Space Time.

A top contender for the most profound insight in the history of physics was a little idea by Galileo 400 years ago.

He realized that speed is relative-in a fundamental way.

Galilean relativity or Galilean invariance states that the laws of physics should be the same "invariant" for all inertial observers.

The laws of physics don't care how fast you're moving.

As long as you aren't accelerating-that's what inertial means.

Drop a ball in a train moving at a constant speed and the ball falls straight down from your perspective, just as it would if you were standing on the train station-again, from your perspective.

This means that you can define any reference frame as having zero velocity and describe all motion with respect to that frame.

There's no preferred notion of stillness in Galilean relativity.

Now galilean relativity was the basis of Isaac Newton's mechanics a half century later, which changed all of physics.

But Galilean relativity had its glitches-for example, it proved inconsistent with electromagnetism as described by Maxwell's equations in the mid-1800's.

Einstein realized that he could fix this inconsistency by just being much more serious about the proposition that laws of physics don't care about velocity.

He realized that even our measurement of the speed of light constitutes a law of physics.

By requiring that all non-accelerating observers measure the same speed of light led him to his theory of special relativity.

In Galilean relativity you can change to a different reference frame just by adding the velocity of that reference frame to all other velocities.

This is called the Galilean transformation.

But that can easily lead to some things having speeds greater than light.

This is fixed in Einsteinian relativity by including this extra factor during that transformation so that nothing ever exceeds light speed.

This is the Lorentz transformation, and the factor that we added is the Lorentz factor.

And the fact that the laws of physics don't change under a Lorentz transformation is called Lorentz symmetry.

A similar idea about the invariance of the laws of physics in accelerating frames of reference versus gravitational fields gave Einstein the general theory of relativity in 1915.

But GR revealed a new contradiction.

Galilean relativity was succeeded by Einsteinian relativity, in part inspired by Galileo's own insight, and motivated in part to fix the contradiction with Maxwell's electromagnetism.

The latter was in turn succeeded by quantum mechanics a decade after general relativity.

But now a new conflict arose-general relativity turned out to be in conflict with the new quantum mechanics, just as Galilean relativity conflicted with electromagnetism.

Efforts to fix this contradiction spawned all of our work towards a theory of quantum gravity-string theory, loop quantum gravity, and many more-and we've covered a lot of these before.

But maybe things don't have to be so complicated-at least, maybe we can get some insight into the answer by doing the same thing that Einstein and Galileo did-by taking relativity seriously.

In relativity there are two fundamental constants-the speed of light "c" and the gravitational constant "G".

With quantum mechanics, a new constant of nature emerged-the Planck constant, h, or h-bar.

When we combine these constants we get various so-called Planck units.

These units tell us where quantum mechanics and general relativity merge-they are the scale at which spacetime is expected to become quantum.

For example, quantum mechanics tells us that probing small distances requires large energies, while GR tells us that the energy curves the fabric of spacetime, and at a certain point the curvature is so great that black holes form.

The distance where this occurs is called the Planck length, 1.6x10^-35 m, and the energy needed to probe that size is the Planck energy-2 billion Joules.

Sounds tricky, but it's not the first time two excellent theories conflicted.

I hope you still remember that Galilean relativity and Newtonian mechanics conflicted with Maxwell's electromagnetism.

And also that Einstein fixed this by doubling down on Galilean relativity-by taking seriously the idea that the laws of physics shouldn't depend on velocity.

So what if we take it even more seriously.

There's another thing that probably shouldn't change depending on how fast you're traveling-and that's the Planck length.

Special relativity tells us that lengths appear shorter along the direction of motion-this is length contraction, and it should apply to the Planck length.

But this length is the scale at which the fabric of spacetime itself takes on strong quantum properties, which sounds pretty fundamental.

So maybe the Planck length shouldn't depend on the velocity of the observer.

That's especially true because it would change only in the direction of motion-implying that the transition between classical and quantum spacetime would depend on direction, which seems weird.

In quantum gravity theories that have some sort of discrete structure underlying the smooth fabric of space, this could mean that structure gets sort of flattened for all but one reference frame.

In general, a contracting Planck length suggests there's a single preferred frame of reference-the one where it's not contracted in one direction.

One more reason to wonder if the Planck length should be invariant- is that it's defined only by fundamental constants G, C, and h-bar.

So what does it even mean for it to shrink?

The Lorentz transformation is a correction to the Galilean transformation that makes the speed of light invariant for all observers.

So what if there's another correction that could make the Planck length invariant for all observers, regardless of velocity?

Well it turns out there is a way to do this.

It was proposed by Italian physicist Giovanni Amelino-Camelia back in 2000, and it's called doubly special relativity, or DSR.

The modern version of DSR is formulated to actually treat the Planck energy as invariant, rather than the Planck length.

The Planck energy is the energy you need to actually probe the Planck length, and is related to it with a simple formula.

If we can keep the Planck energy independent of velocity then we've done the same with the Planck length.

Now because we're now talking in energies, we get to use Einstein's equation relating energy to mass and motion.

You've definitely seen part of it: E=mc^2-probably the most famous equation in the world.

The full version is this, It says that a particle's energy is equal to the energy bound up in its mass plus the energy of its motion, expressed here in terms of the particle's momentum.

It's also called the relativistic dispersion relation-which I'm telling you because I have to keep calling it the "dispersion relation" for the next bit.

Now remember the Lorentz factor was added to the Galilean transformation in a way that only makes it significant close to the speed of light, we can also add this factor to the dispersion relation that only becomes relevant at energies close to the Planck energy.

This is the modified dispersion relation-the MDR.

That extra bit stands for one or more terms that include the ratio of particle energy to the Planck energy.

For energies much lower than the Planck energy-which means for most of the universe-this part becomes close to zero and so we just have the original dispersion relation.

Just like how the Lorentz transformation becomes the Galilean transformation for velocities much smaller than that of light.

The higher the power n, the closer you need to get to the Planck energy to observe the effect, and the coefficient eta also governs the overall strength of the term.

If those terms in the MDR become significant compared to the original terms of the dispersion relation it would mean that energy and momentum conservation as we know them no longer hold.

See these conservation laws are a direct result of the Lorentz symmetry, via Noether's theorem-something we covered a while back.

If Lorentz symmetry is broken then energy and momentum conservation also break.

Which would be fine -we'd end up with a more fundamental symmetry and more fundamental conserved quantities to go with it.

But energy and momentum conservation have been pretty fundamental to the development of all of physics.

For example, we use these laws to determine how particle interactions happen in quantum physics.

They don't just determine how particles exchange energy and momentum-they determine what interactions are even possible.

And that gives us a way to potentially test the ideas of doubly special relativity.

Here's an example.

A neat trick that the massless photon can do is to turn itself into a pair of massive particles-one of matter and one of antimatter.

For example, a photon could become an electron and positron.

However, due to the normal rules of energy and momentum conservation, this "pair production" is not possible for a single photon without help.

Pair production only happens when a photon interacts with something else-often another photon.

But if energy and momentum conservation is broken due to our modification of the dispersion relation then it becomes possible for a single photon to undergo pair production.

That would be a huge deal.

Currently, photons and other massless bosons like gluons are thought to be completely stable in isolation.

If they aren't really stable then that's something we could potentially test.

It would take a very energetic photon-remember, the modified dispersion relation starts to act differently approaching the Planck energy.

But this non-Lorentzian behavior can still happen the further from the Planck energy, albeit with less likelihood.

If some form of the modified dispersion relation is right, then there should be a faint possibility that a photon of an extremely high but not impossibly high energy will decay via pair production given a long enough time.

And if any one such photon is given enough time, then that small probability becomes a large one.

That means there should be a limit to how far a high energy photon should be able to travel through space before it decays.

We have seen very high energy photons from objects pretty far away-especially from supernova remnants in the Milky Way Galaxy.

Apparently though there was not a high enough chance of decay of these photons by pair production to eliminate them on their way to us.

Now another reaction forbidden by normal conservation laws is for an electron to spontaneously emit a photon without first interacting with something else.

This "vacuum cerenkov radiation" shouldn't exist, but with a modified dispersion relation it just might.

But we have not seen any evidence of vacuum cherenkov radiation in astrophysical sources.

Between the absence of pair-production decay and vacuum Cherenkov radiation, researchers have been able to constrain the strength of any possible modified part of the dispersion relation.

So apparently we need a more senstive test.

Another hotly debated prediction from some DSR theories is that the speed of light might actually be different depending on its wavelength, and hence energy.

But why should fixing the Planck length or Planck energy cause the speed of light to vary?

Well, one way to express the speed of light-or anything really-is as the ratio of kinetic energy to momentum-and that's c for light.

But in our modified dispersion relation, energy has these extra bits to it.

At very high energies those extra bits become important and cause the energy-momentum ratio to deviate from c. So if this ratio actually defines the speed of light, then we have a way to change that speed at extreme energies.

Whether high energy photons move faster or slower than low energy ones depends on the nature of the modified dispersion relation being considered, but either way this can potentially be tested.

Now scientists have been looking for tiny energy-dependent variations in the speed of light for a long time.

We've even talked about it before.

One way to do this is to look for different arrival times of photons from the same distant astrophysical event.

For example, gamma-ray bursts are a class of uber-bright supernova that produce high-energy gamma-ray photons as well as photons at all other lower energies.

We know that all these photons all started their journey to us at the same time-the time of the explosion.

So in principle we should be able to tell if the higher energy ones traveled at a different speeds-they'd arrive at a different time.

Perhaps earlier, perhaps later.

The most recent and highest energy ever detection of such an event, is gamma-ray burst 221009A, comes from the Large High Altitude Air Shower Observatory (LHAASO) in 2022.

No difference in arrival time of high versus low-energy photons was detected within the rather high sensitivity of this experiment.

That allowed us to place pretty tight bounds on any energy variability of the speed of light.

And that in turn places limits on the strength of the extra terms in our modified dispersion relation.

The researchers claim to have 95% confidence that if there were linear corrections to the speed of light, they'd have to be at energies quite a bit larger than the Planck energy.

This effectively rules out the strongest corrections-those where the power-the n in our MDR-equals one.

Quadratic corrections-where n=2--have to be at least on the energy scale of 10^11 GeV.

That's around 5 orders of magnitude more energetic than the most energetic gamma ray ever detected.

It's not so surprising that current experiments haven't detected any of the predictions of doubly special relativity or found evidence of the modified dispersion relation.

As far as we know, E still equals mc^2.

But in this case we haven't yet proved Einstein right.

The potential breakdown of Lorentz symmetry at the Planck scale is still an issue that we don't understand fully, and it may be that the alterations needed to the dispersion relation are just too small for current methods or current space explosions.

But at least we know how to look for these effects, which has been a real challenge for anything in the realm of quantum gravity.

And it sure would be cute if the first clue to the quantum nature of space came from the 400 year old insight by Galileo.

No matter how fast you go, physics looks the same-even the length-scale of Planckian spacetime.