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.
Hey, Everyone! You know how black holes pull everything in with irresistible gravity? And you
know how the upcoming holiday season often feels the exact same way? Well that’s why we’d like to
welcome you to our first ever Black Hole Friday Sale! On Friday, November 29th, for one day only
the entire Merch store catalog will be on sale for 20% off. We also have a bunch of brand new merch
such as the first ever Multifunctional Space Time Pen, which features a level, rulers in inches,
centimeters, and millimeters, a stylus and micro screwdriver. It’s basically a low-grade sonic
screwdriver. Stand by for the full-tech version. If you need somewhere to use that pen, We also
have a new Standard Model LED night light for when you’re working on your theories
late into the evening, a new Neural Net desktop mat to keep your computer stylish while your run
your simulations, a 520 Piece Puzzle for when you’re waiting on those simulations to render,
our first ever Poster that will remind you of where all elements came from, and a Penrose
Diagram Black Hole Rocks Glass, for when you need a tasty beverage to celebrate of your successful
theory. Can we legally say buying Space Time merch will help you formulate a Nobel Prize
winning physics theory? That’s a grey area. But we can offer you 20% off every t-shirt, hoodie,
patch, pin we’ve sold this year during their special re-appearance on the Merch store. which
you could theoretically wear to holiday events or to accept any kind of important international
prize. And speaking of the non-Americans, if you're one of our many international viewers,
November 29th will be that last date we can guarantee international delivery before December
25th. So if you need a gift for any special person in the galaxy, including yourself, remember to go
to PBSspacetime.com/shop on November 29th for our 1st ever Black Hole Friday sale!