Monday, January 04, 2010

Space is very fine-grained

It would take you a lot longer to hike a significant distance over very hilly terrain than it would over a completely flat plain. For much the same reason, it would take light longer to cover the same distance depending whether the space through which it moves does or doesn't have large "hills".

But what does it mean for space to contain "hills"? And how large do "hills" need to be to make a difference?

Consider the second question first. There's no natural place on Earth that is perfectly flat, of course. So if you look closely enough, there are always "hills" of some size to cross when you're on a hike. But if the height of those hills is a lot less than the length of your stride, they will make little difference. In the same way, if the irregularities in the texture of space are much smaller than the wavelength of light, they won't make much difference either.

Physicists aren't even sure that space does contain "irregularities" at very short distances. No measurement yet made has given any evidence of irregularities. But all theories of quantum gravity – none of which is yet considered satisfactory – predict that irregularities must exist at a sufficiently small scale.

No attempted quantum theories of gravity are satisfactory yet, because physicists have not been able, with a mathematically consistent theory, to reconcile general relativity and quantum mechanics at very small scales. Nature, however, is somehow able to do the trick. So it seems quite plausible that space does have irregularities at sufficiently small scales. The only real question is "how small?"

Recent research now says, "smaller than can presently be detected."

And how would one go about detecting the smallest irregularities? With light whose wavelength is about the same size, of course. Which would mean photons with the highest energy one can find. So far, photons with the highest energy physicists know of are not indicating any quantum irregularities in space.

We're talking about photons with energies far higher than can be produced in laboratories. The most energetic photons correspond to the part of the electromagnetic spectrum known as gamma rays. By convention, physicists regard gamma rays as electromagnetic radiation with a wavelength of less than 10-11 m (10 picometers). That corresponds to an energy of 105 eV (eV = electron-volt). (Recall that photon energy is related to frequency by the relation E=ℎ&nu, where ℎ is Planck's constant and ν is the photon frequency. Since wavelength λ is inversely related to ν so is energy; i. e. E ∝ 1/λ.)

Gamma rays are produced naturally in some types of atomic decay, and such gamma rays have energies up to 107 eV. That's not terribly energetic in the great scheme of things. The Large Hadron Collider will be able to accelerate protons up to energies around 1013 eV.

Since photons have no electric charge, unlike protons, they can't be accelerated the way protons can. Much more energetic gamma rays can be produced in particle-antiparticle annihilations, but such high-energy photons are rather rare, even in cosmic events like gamma-ray bursts, which may produce gamma-rays with energies of 3×1010 eV or more (3.34×1010 eV is the largest yet observed). That's 3×105 times as energetic as the least energetic gamma rays and so corresponds to wavelengths ~.3×10-16 m.

Fortunately, even with gamma rays in this energy range, much smaller irregularities in space can be detected if the gamma ray photons travel a very large distance. Suppose, for instance, that a gamma-ray photon were detectably slowed down only by 1 part in a million, a factor of 10-6, over a distance of 100,000 light-years, about the diameter of a large spiral galaxy.

But really cosmic distances at which we can observe gamma-ray bursts are about 1010 light-years, or 105 times the diameter of a galaxy. Since a light-year is about 1016 m, we're talking scales like 1026 m. On that scale, the possible slow-down of a gamma-ray photon could be very noticeable – roughly 1 part in 10.

The result is that if we can detect how much a gamma-ray photon is slowed down over a distances around 1010 light-years, we could be able to probe energies much higher than those possessed by a gamma-ray photon itself, and therefore distance scales much smaller than a very energetic gamma-ray photon wavelength of about 10-17 m.

Detecting small differences in the velocities of photons of different wavelengths is made much easier when both photons travel 1010 light-years over a time of (naturally) 1010 years. Gamma-ray bursts, fortunately, produce gamma rays over a range of energies that differ by several times 105 from smallest to largest.

To be more concrete, suppose you have a gamma-ray burst at a redshift of z~.9. That corresponds to a distance of ~7×109 light-years and a travel time of ~7×109 years. The wavelength of photons traveling that distance is stretched by a factor of 1.9 (i. e. 1+z), but there's still a factor of several times 105 between the wavelenghts of the most and least energetic gamma-ray photons, so they are readily distinguishable. If the difference in arrival time of gamma-ray photons is 1 second, that is only one part in 2×1017 (i. e. 7×109 years times about 3×107 seconds/year ≅ 2×1017 seconds).

It's reasonable to suppose that the slowdown of the least energetic gamma-ray photons due to irregularities in space is negligible. Suppose we could relate the slow-down of the most energetic photons to the actual size of spatial irregularities. For instance, suppose a spatial irregularity of 1 part in 1017 of the photon's wavelength caused a slow-down that was also 1 part in 1017 of the photon's velocity. That would cause a 2 second delay in photon arrival times – which is very readily detectable.

Recall that energetic gamma-ray photons from gamma-ray bursts have wavelengths of .3×10-16 m or less. Thus we might expect to be able to detect spatial irregularities as small as ~.3×10-33 m = 3×10-34 m = 3×10-32 cm.

In fact, it is now possible to measure arrival times of photons from gamma-ray bursts to within mere hundredths of a second. So we can probe length scales around 10-33 cm. Interestingly enough, this is very close to the Planck length scale lPlanck ≈ 1.62×10-33 cm.

That's just a back-of-the-envelope calculation based on hypothetical data. But we don't need to be hypothetical, because photons from the gamma-ray burst GRB 090510, which was observed on May 10, 2009, were measured very precisely. The most energetic photon observed had an energy of ~3.1×1010 eV (31 GeV), and it showed up precisely .829 seconds after the very first photons from the burst were detected. Further, spectroscopic observations of the afterglow from this burst showed a redshift of z very close to .9, as in our hypothetical example.

Since we don't have a reliable quantum gravity theory, we don't know exactly how much of a slowdown very high-energy, short-wavelength photons should experience. However, several theories predict that, to first order approximation, if vph is the effective average velocity of the photon, then its ratio to c, the speed of light, should satisfy |vph/c - 1| ≈ Eph/(MQGc2). One can think of MQG as the "mass" that the photon would have in the quantum gravity theory so that the photon energy is c2 times the mass.

Given that notation, then if you have two photons that differ in energy by ΔE, the difference in arrival times should be Δt ≈ (|ΔE|/(MQGc2))D/c, where D is the distance traveled. In this relationship, all quantities except for MQG are directly measured, which implies a value of MQG.

The most sensible unit in which to measure MQG is in terms of the Planck mass, MPlanck ≅ 2.17644×10-5 g, which is quite a lot for small things, being about the mass of 20 million bacteria.

Most quantum gravity theories predict MQG ≤ MPlanck, so that the ratio MQG/MPlanck ≤ 1. Surprisingly, however, measurements made of GRB 090510 imply that this ratio is actually no smaller than 1.2, and could be quite a bit larger – as much as 100 or so.

There's a great deal of uncertainty in the estimate of Δt, the difference in arrival times between the 31 GeV photon and lower energy photons that were emitted at the same instant. That's because there is no way to know how long after the start of the GRB event the 31 GeV photon was emitted. Since it was observed .829 seconds after the very first photons, a large part of that delay could actually be due to emission of the 31 GeV photon at any point up to .829 seconds after the start. That would make the actual Δt much smaller, and MQG much larger.

Only the most conservative assumption, with the 31 GeV photon emitted as early as possible, gives MQG/MPlanck ≈ 1.2. More realistic assumptions would make the ratio 100 or more.

In any case, all of these estimates "strongly disfavor" the simplest theories of quantum gravity, in the words of the research paper describing the observations. Otherwise said, spacetime at the smallest scale must apparently be much less bumpy than most theories predict.

Here's the research paper and abstract:

A limit on the variation of the speed of light arising from quantum gravity effects (11/19/09)
A cornerstone of Einstein's special relativity is Lorentz invariance—the postulate that all observers measure exactly the same speed of light in vacuum, independent of photon-energy. While special relativity assumes that there is no fundamental length-scale associated with such invariance, there is a fundamental scale (the Planck scale, lPlanck ≈ 1.6×10-33 cm or EPlanck = MPlanckc2 ≈ 1.22×1019 GeV), at which quantum effects are expected to strongly affect the nature of space–time. There is great interest in the (not yet validated) idea that Lorentz invariance might break near the Planck scale. A key test of such violation of Lorentz invariance is a possible variation of photon speed with energy. Even a tiny variation in photon speed, when accumulated over cosmological light-travel times, may be revealed by observing sharp features in gamma-ray burst (GRB) light-curves. Here we report the detection of emission up to ~31 GeV from the distant and short GRB 090510. We find no evidence for the violation of Lorentz invariance, and place a lower limit of 1.2EPlanck on the scale of a linear energy dependence (or an inverse wavelength dependence), subject to reasonable assumptions about the emission (equivalently we have an upper limit of lPlanck/1.2 on the length scale of the effect). Our results disfavour quantum-gravity theories in which the quantum nature of space–time on a very small scale linearly alters the speed of light.

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Further reading:

Gamma-ray Observations Shrink Known Grain Size Of Spacetime (10/28/09) – Science News

Fermi Telescope Caps First Year With Glimpse of Space-Time (10/28/09) – NASA/JPL

7.3 Billion Years Later, Einstein’s Theory Prevails (10/28/09) – New York Times

Gamma-ray burst restricts ways to beat Einstein’s relativity (10/28/09) – Symmetry Magazine

Special relativity passes key test (10/28/09) – Physics World

Gamma-ray photon race ends in dead heat; Einstein wins this round (10/28/09) –

An intergalactic race in space and time (10/28/09) – Nature

Astrophysics: Burst of support for relativity (11/19/09) – Nature

Space-Time Observations Find Einstein Still Rules (10/28/09) –

Quantum gravity theories wiped out by a gamma ray burst (10/28/09) – Ars Technica

A Gamma Ray Race Through the Fabric of Space-Time Proves Einstein Right (10/29/09) – Discover

Nature, NYT report the demise of Lorentz-violating theories (10/29/09) – The Reference Frame

Einstein Still Rules, Says Fermi Telescope Team (10/28/09) – Universe Today

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