faster than a speeding photon

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14/12 A Neighborhood of Infinity: Faster than a speeding photon

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A Neighborhood of Infinity

Saturday, April 04, 2009

Faster than a speeding photon

How to outrun a photon

I thought it would be fun to try to give a readable account of Unruheffect. It's a surprising phenomenon, and there isn't universal

agreement over what exactly the theory predicts, let alone whether theeffect has ever been observed. It has important implications for physicsand philosophy and may even give a way to test some aspects ofquantum gravity in the lab.

One way to start the story is consideration of this problem: if a photonis speeding towards you, can you outrun it? Let's simplify things a bitso that we're considering motion in one dimension.

If we're confined to one dimension, we can't dodge the photon, we can

only hope to remain ahead of it. As the only things that can travel at thespeed of light, c, are massless things like photons, it seems that there isno hope for a massive thing like a person in a spaceship to avoid it. Thephoton will always be faster than you, and so it'll catch you.

But in theory you can outrun a photon! Do you see the flaw in the abovereasoning that made it seem impossible?

The best way to make things clear is to draw a diagram. We'll plotsome graphs of position vs. time for some photons and spaceships.

We'll have time going up the vertical axis and position along thehorizontal axis. Here's an example:


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I've chosen units so that one second on the vertical axis is drawn thesame size as one light-second on the horizontal axis. The net result isthat photons always travel at 45 degree angles to the axes. Massiveobjects, that travel slower than light, are confined to travel on coursesthat have angles of smaller than 45 degrees with respect to the vertical

axis. The path of the photon is the diagonal black line and the path of aspaceship is in red. It starts to the right of the photon but as we moveup the time axis the photon eventually catches up with it.

If the spaceship travels faster then it will follow an angle closer to 45degrees. Here are a pair of paths corresponding to faster spaceships:


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The faster the ship is, the further it gets before the photon catches up.But we're just putting off the inevitable. It seems that whatever we do,the photon will always catch up.

But there's a hidden assumption in the above. By drawing straight lines

for the spaceship I was assuming it was travelling at a constant velocity.But there's no reason for that to be true. Here's a different path thespaceship could follow:


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At no point does the red path of the spaceship meet the black path ofthe photon. And yet at no point does the red path reach 45 degrees tothe vertical axis. In other words, the spaceship never travels at thespeed of light, and yet the photon never catches up with it. Spaceships

can outrun photons!

So what kind of path is that? It's actually a hyperbola and itcorresponds to a spaceship accelerating at a constant rate. You might

wonder how it can be constant acceleration when the speed of thespaceship never exceeds that of light. From an external observer's pointof view, after a while it does look like the ship is travelling at a more orless constant velocity close to the speed of light. But from the point of

view of someone on the spaceship it feels exactly like constantacceleration. So that is the path that would be taken by a spaceship

with its thrusters firing at a constant rate.

An event horizon!

I chose that path so that the spaceship stays just in front of the photon.A photon that starts slightly to the right will eventually catch up withthe ship. But photons starting further to the left of the ship will neverreach it. This means that absolutely nothing starting to the left of the


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horizon. But that's just the start of the weirdness. When we throwQuantum Mechanics into the mix something much weirder happens.

Matter from a vacuum

It's well known that physicists expect black holes to emit particles asHawking radiation. But our accelerating observer sees something like a

black hole, so we might expect them to see something like Hawkingradiation. We also know that an observer at rest sees no event horizon.

Which means that we might predict that accelerating observers seeparticles that observers at rest don't. Can we take such a predictionseriously?

Let's look a bit more closely at this. According to a popular view ofquantum mechanics, the vacuum is teeming with vacuum fluctuations -ephemeral particle-antiparticle pairs that briefly come into existence

and then annihilate each other. In the diagram below I've drawn someof these events:

As we follow up the time axis, pairs of (complementary coloured)particles come into existence and then annihilate each other. Theseevents are so fleeting that they have no effect on our particle detectorsand we see a vacuum. But note that I've drawn one of these events


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straddling our apparent event horizon. From the point of view of anaccelerating observer this looks like a pair of particles coming intoexistence, but because of the argument I sketched above, they seem tofreeze near the event horizon. In other words, to an acceleratingobserver these fleeting events are no longer fleeting, they look like realparticles coming into existence and sticking around forever.

Accelerating observers appear to see particles in a vacuum!

What I've described above is absolutelynota rigourous argument. Butamazingly, when you use the machinery of quantum field theory youend up making exactly the same prediction: accelerating observers seeparticles. This is known as the Unruh effect. When you do this properly

you can compute a bit more detail. It turns out that the energies of theparticles are random with exactly the same distribution asblack bodyradiation. In other words, the vacuum looks like it has a glowcorresponding to a particular temperature that is proportional to theacceleration. But it's not a bright glow. You need to accelerate at about

1020 m/s2 before the temperature appears to be 1K. Building athermometer that can survive such accelerations is no mean feat. So itlooks like the Unruh effect is a curiosity that might never be observedin the lab.

But it has been suggested that the Unruh effect has already beenobserved. There aren't many things that can survive that kind ofacceleration, but an electron can, and an electron can behave like athermometer. Electrons in circular particle accelerators routinelyundergo the kinds of accelerations we're talking about. They do so

because they are driven by a magnetic field. Now an electron has spin,

so you can think of it as a bit like a little electric current running roundin a loop. That means an electron is like a little dipole electromagnet.Magnets in magnetic fields tend to want to line up along the field -that's how a compass works. So electrons that spend long enough in aparticle accelerator, eg. those in a storage ring, should eventually lineup with the field. Lining up like this is known as polarisation, and inthis particular case it's known as he Sokolov Ternoveffect. But when

we look at electrons in a storage ring it turns out they're not quitecompletely lined up, they're slightly depolarised. This is easilyexplained by Unruh radiation - they're constantly accelerating and so

they feel themselves to be in a hot environment. The continualinteraction with this hot environment causes the electron spins to be a

bit randomised, so they don't all line up nicely.

Unfortunately this isn't definitive evidence for Unruh radiation becausewhen we carry out the full calculation of the Sokolov-Ternov effect itturns out that it predicts partial depolarisation anyway. Now it lookslike we don't have evidence for the Unruh effect. But it's not thatsimple. The Unruh effect isn't a new effect made up by a physicist. It's aprediction based on a new way of looking at fairly conventional physics.


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31 comments:

Fergal said...

Is there a minimum rate of acceleration required to produce this effect? I

presume different rates give different asymptotes and you need one that beats45 degrees.

How does this minimum acceleration relate to the acceleration produced by ablack hole at some radius? Is there some nice relationship involving theSchwarzchild radius?

Saturday, 04 April, 2009

sigfpe said...

Every acceleration eventually results in a 45 degree angle. The speed limit ofnature is the speed of light and so that speed always provides the asymptote.

BTW I generated that curve in NodeBox using the parametric formula (cosht,sinh t)

You can relate this to a black hole by thinking about hovering over a large

black hole. Locally the gravitational field looks like a constant vector, likehow on the surface of the earth we can use the vector (0,-g,0). As you hover,

you feel gravity pull you down onto the floor of your spaceship. This feelsexactly like accelerating at a constant rate. According to Einstein'sequivalence principle, the feeling of gravitational pull is the same thing as thefeeling of acceleration. And so you'd expect a constant accelerator to see thesame thing as a black hole hoverer. One sees Hawking radiation (whichresults in a particular temperature) and so the other must see the same thing(in which case it's called the Unruh effect, and should result in the sametemperature).

The relevant formulae are on Wikipedia so it should be easy to see if the twoapproaches give the same formula.


Brandan said...This comment has been removed by the author.Saturday, 04 April, 2009

Brandan said...


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hi. i'm not really all that knowledgeable mathematically, and i'm a bitconfused by your writing. perhaps you could elucidate, if you feel so inclined.:)

you mention thinking of things in one dimension, yet you have demonstratedthe point on a 2-dimensional chart. how does something imagined in onedimension possess the capacity for 'direction'?

i want to read the rest, but i'm barely stumbling atop that, waiting for themoment i lose my footing and crash into the waters of vitriolic indifference.

i'm sorry for the repost, but i misspelled 'lose' as 'loose' and felt immenselystupid.


sigfpe said...

Brandan,

I'm talking about one dimension of space and one of time. So space looks one-dimensional to people in it, but spacetime is two-dimensional.

Of course this is just for simplification. In reality the theory is worked out ina 4D spacetime with three dimensions of space and one for time.


Brandan said...

oh yeah, a simplification! thanks for the clarification. i tend to think of time asbeing an abstract recognition of the evolution of matter rather than aconcrete thing. time being a manifestation of the capacity for memory and allthe prefrontal cortex's various functions. i always forget about time.


carlosscheidegger said...

This is from the wikipedia page: The Unruh effect, described in 1976 by BillUnruh of the University of British Columbia, is the prediction that anaccelerating observer will observe black-body radiation where an inertialobserver would observe none. In other words, the background appears to bewarm from an accelerating reference frame.

So could we interpret the CMB radiation as some sort of acceleration? That'salso blackbody...



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Andrej Bauer said...

Thank you for a very entertaining reading by my morning coffee! However,where are the monads? -) I expect your next post to contain haskell monadsfor relativity and quantum mechanics. You know, Feynman diagramsliberated from the chains of fortran and the like.

Sunday, 05 April, 2009

Anonymous said...

This is interesting. Could this effect explain the cosmic background radiation?If you need 10^20 m/s^2 to get 1K, then what do you need to get 3K? I readthat the expansion of the universe is acclerating (dark energy/matter etc) so ifthe fabric of space at the boundary of our visible universe was acceleratingaway so fast, we'd see (a) CMB right? I daresay an acceleration of>10^20mps2 would be quite destructive though.... let's see, 10^20mps2 /15Gly ~ 1 um/s^2 per metre. Follow? If each metre were to stretch by 1

um/s^2, then the effect 16Gly distant at the edge of the visible universe wouldamount to 10^20m/s2. Hmm, something tells me 1um/s2 would be enough topull galaxies apart...

[a different] Fergal.


Dan Doel said...

I don't really see how the explanation in the first part could be applied to

electrons in a circular accelerator. Am I correct in thinking that theacceleration is that which is required to keep them traveling in a circularpath? If so, then though the magnitude of the acceleration is constant, thedirection is continually changing, so the situation seems rather different thanthe 1-dimensional case, or a space ship firing its thrusters at a constant rate.


sigfpe said...

Anonymous,

I believe this paper is on what you're asking about, particle creation in anexpanding universe: http://adsabs.harvard.edu/abs/1968PhRvL..21..562P It'smentioned here:http://www.dartmouth.edu/~physics/news/colloquium.archives/wald_11_3_06.pd


sigfpe said...


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Dan Doel,

The papers on circular accelerators discuss the difficulty of applying thetheory to circular motion and at least one paper says it doesn't arise incircular motion.

But for a large enough circle, the motion will look like approximately linearmotion for short intervals at a time. So I'd expect the Unruh effect to look

similar.


Anonymous said...

Hi, it's [the 2nd] Fergal again,

Well, if you take 1 um/s^2 per metre, and apply that expansion rate to all themetres between us and the centre of the earth (6000km), you get 6m/s^2,

just a little less than g. On the other hand, this effect completely outweighsthe acceleration due to the sun (2e30kg, 150e9m -> 6mm/s^2 acc, Vs 1e-6 *150e9 = 150000m/s^2 total expansion) if true therefore, the earth would flyoff with the rushing expansion of space, unless if G was actually a much biggernumber than previously thought - i.e. we'd have to rethink gravitation.

Much easier is to assume that space just expands in the vicinity of darkenergy/matter, which, fortunately, is not here.

Fergal.


Aaron McDaid said...

Dan, sigfpe,But with a big enough circle, the acceleration towards the centre of theaccelerator is small and we won't see any Unruh effect. Unless there is to be

very high angular acceleration to compensate. Don't we want to see very highacceleration in a constant direction, even if only for a short time?

Monday, 06 April, 2009

sigfpe said...

Aaron,

Here's an example of a storage ring with some figures:http://www.lns.cornell.edu/public/lab-info/cesr.html

A big circle and high acceleration!


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BTW I'm not implying that you require constant acceleration for the Unruheffect to appear. I'm saying that if you want to use the linear accelerationformula with confidence to predict the amount of Unruh effect from circularacceleration you want approximately constant acceleration for reasonablylong periods of time. There's another calculation you could carry outspecifically for circular motion - but it'll be more difficult. Similarly Isketched out what happens for constant acceleration for all time but you

could also do the calculation for something that starts at rest, accelerates fora bit, and then remains at a constant velocity. Again, it's much harder to workout the details, but you still expect to see the effect.


Aaron McDaid said...

When I said "we won't see any Unruh effect" I meant to say that it might bejust as hard to see in a big ring as in a small one, not that the Unruh effect

disappears altogether.

For a given speed, say 90% of c, an increased radius will make theacceleration more like linear acceleration, but the magnitude of theacceleration must be smaller.

But, to be honest, I don't really know much about how particle acceleratorsreally work so I should have kept my mouth shut :-) I think I understand theprinciple, but I should stop thinking I can help design a practical experiment:-)


Josefsaid...

This is a really silly question but is there any particular reason you put timeon the vertical axis instead of the more conventional horizontal axis?

Tuesday, 07 April, 2009

Charlie said...

I don't think josefs question is silly. It's common practice and makes moresense to see position as a function of time (f(x) = y). And how can theacceleration be constant if the velocity never get's greater than c. Yourediagram shows that the accelelration decreases infinetly approaching zero. Idon't see how this would be significantly diffrent from a spacehip moving atconstant velocity very close to the speed of light. So my question is: What isthe relevance of acceleration to observing this unruh effect, other thanacceleration is required to reach velocities close to the speed of light?



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sigfpe said...

Charlie,

Although to an external observer it looks like the red path is just approachinga constant velocity it turns out that in the frame of reference of somethingmoving along the red path it feels like constant acceleration. This is not anobvious thing. You can relate it to time dilation. To someone on the

accelerating path it feels like they're zipping along much faster than how itlooks from the outside. There's some discussion here:http://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity)


sigfpe said...

Charlie,

You ask a good question. Note the representative little loops I drew torepresent particle-antiparticle creation-annihilation. These are just a"classical" picture of something quantum mechanical and so the metaphorsare straining a bit at the edges. But roughly speaking: you can think of theuniverse as randomly distributed with lots of these little loops. They aredistributed in such a way that the overall distribution looks identical to anobserver flying by at constant velocity, no matter what that constant velocityis. It's only when acceleration occurs that this distribution starts lookingdifferent. But to go any further requires doing some real quantum fieldtheory...


Charlie said...

Yes, that's right! I completly overlooked the fact that velocities aproaching cdistort the very basic stuff like mass, time and lenght. It makes perfect senseto me now. I sincerely apologize for posting stupid questions like that.


sigfpe said...

Andrej,

I was thinking about a black hole monad, related to myheat monad. :-)

Wednesday, 08 April, 2009

genneth said...

Aha! Finally something on your blog that I understand!


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There is also an entirely classical way to view the Unruh effect: Doppler shift.Imagine you are accelerating into a wave of fixed frequency due to Dopplershift, you will see it at all sorts of different frequencies. If you do the maths (aFourier transform) and average over all frequencies, you recover the Planckdistribution for a field at the Unruh temperature. If you're familiar with theQFT formalism, then you're simply taking the Fourier transform of thecorrelation function along the desired trajectory.

In the circular case, you do get a similar effect, but the distribution is notPlanckian. To a physicist, thermal is really a statement about time-independence of a statistical state. There are quite a few different trajectories

which all give a thermal distribution, and all of which can be assigned atemperature in some appropriate sense. In fact, recall that a constantacceleration is really just a constant rotation...

A quick plug: my own work has been focussed on whether it's possible to seethe Unruh effect in atomic Bose-Einstein condensates. We are hoping to move

a detector (atom in various electronic states) in a circle in a BEC. Thephonons in the BEC act as a quantum vacuum, and should give rise to ananalogous effect. Hopefully in the next couple of years we can get someexperimental groups to really take a crack at this.


sigfpe said...

genneth,

I was thinking of giving an account based on FFTs but I wanted to try to getsome talk of creation and annihilation operators in there and couldn't figureout a "pop science" explanation of them.

Good look on the BEC experiments. I seem to remember reading aboutsomething similar with researchers looking for Hawking radiation near a BEC"event horizon".


Sundar said...

"But for a large enough circle, the motion will look like approximately linearmotion for short intervals at a time"

But then, if you are approximating the circular motion as a linear one, don'tyou lose all the acceleration there? Because by definition, the onlyacceleration in a circular motion is the change of direction. Once you lose thatand assume linear motion, you're just travelling in constant velocity, isn't itso?


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I understand more rigorous analyses have been performed regardingextending this to circular motion, but it's the 'simplification' here that appears

wrong to me...


Anonymous said...

Sundar, what you are thinking of is constant circular motion, with no angularacceleration. Electrons in a particle accelerator certainly have acceleration inthe direction of motion (this just requires a greater force to maintain thecircular path).

At least, I think that's what you were talking about.

If the discussion is constant circular motion, then shouldn't the falsecentrifugal force provide constant acceleration? However, we do have the

problem with our light-beam analysis (light wouldn't travel with the circle).This might not be a problem though, if we let the centripetal force begravitational, the orbit keep the spaceship from falling (instead of thrusters),and just consider light as radiating from the center. I don't know if Iunderstood this anywhere close to correctly.

Tuesday, 05 May, 2009

Clarksaid...

It would be interesting while thinking of a proton as a singularity. To produce

controllable black holes that can be manipulated to travel in a direction wechoose.

The next part is building a ship that can safely orbit said miniature black holeand manipulate its direction.

Adding this to my to do list.

Wednesday, 06 January, 2010

doug said...

I believe that you are mistaken. The photon would only appear to never catchthe ship from the reference frame of whatever the ship is accelerating withrespect to. From the reference frame of the ship, the photon would stillappear to approach at the speed of light and pass the ship. That's just how fastlight travels.


sigfpe said...


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You're not really outrunning the photon, because you're not going faster thanit - instead you're keeping it at a steady state arms length.

I don't know where you're getting a finite craft with infinite energy (unlessyour sucking it in the nose of the craft!) or if that craft was an acceleratinguniverse popping into another infinitely larger accelerating universe...

Hubble tells us our universe is accelerating, and that we may be that craftthough... but doesn't tell us what exactly this accelerating force is. What theforce people call dark energy is.

Some think it's eternal inflation, others Einstein's cosmological constant,others yet again have labeled it quantum gravity, or now the latest fadeveryone loves to call the entropic force.

Whatever it is, may the force be with you! ^)

Do you have a favourite?



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faster than a speeding photon

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