energy – Energy and Time

Switzerland day 42: Bern & Einstein

Bern was where Albert Einstein was working as a patent clerk during his “miracle year” of 1905. In 6 months, he published 4 papers any one of which should have been worth a Nobel prize:

Photoelectric effect (submitted March 18, published June 9)
Brownian Motion (submitted May 11, published June 19)
Special Relativity (submitted June 30, published September 26)
Energy = Mass (submitted September 27, published November 21)

It’s probably the most productive year any scientist has ever had. None of the math was terribly hard, but each paper revolutionized an entire subject by looking at it differently.

September 30th: First target of the day was the Einstein Museum.

I got off to a rocky start in the museum, which restated a very wrong but perniciously widespread misconception.

This is wrong because (1) matter/mass/energy all bend space-time (not just space), and (2) time is always bent at least as much as space. Around the Earth, for slow-moving objects, gravity is more than 99.9999% due to bending of time, and less than 0.0001% due to bending of space. For extremely fast-moving objects it get closer to 50-50. So simplifying to “matter bends time” is OK and sometimes approximately correct, but simplifying to “matter bends space” is always wrong. Yet everyone does it. Jim Al-Khalili. Stephen Hawking.

Other than that, the physics was pretty good. However, a significant chunk of the exhibit was devoted to the rise of the Nazis and the development and use of nuclear weapons. I wasn’t expecting it to be so depressing.

Some random buildings in Bern:

We had lunch at the Äss Bar, a discount restaurant that gets 1-day-old bread from nearby bakeries and makes cheap sandwiches from it.

A famous clock tower was right on our way, so we waited for it to strike, but it was underwhelming. A few figures to the right of the red dial moved.

Farther down the same street was our second goal for the day: Kramgasse No. 49, the “Einsteinhaus”, the apartment where Einstein lived in 1903-5.

If you lean out Einstein’s window and look left, you can see the clock tower a half a block away. The theory of relativity is based on the behavior of “clocks” and “rulers”; I think I know what inspired the clock part. 🙂

Then we walked around town a bit.

Finally we had to head out for our drive to the Swiss Alps. They became obvious long before we reached them.

Energy and Time, Chapter 1: Deja Vu

Deja Vu (March 2009)

It all started innocently enough. I had been auditing the upper division Quantum Mechanics course at CSU to try to bring my QM skills up so I could get better at Quantum Computing. It was fairly relaxed; I wasn’t doing any of the homework, but I took detailed notes in TeX, complete with formatting the equations on the fly. Mark Bradley, the teacher, was a dynamo at the blackboard though, so it was often a struggle to keep up with his rapidly scribbled equations. So I have very pretty but sometimes incomplete notes.

And then one day in late March, I noticed something a little odd. Or rather, I noticed something that should have been familiar and comfortable and old hat to me, only now it seemed odd and suggestive. The electron wave functions in stationary solutions to the Schrodinger equation . . . you know, things like ”atomic orbitals” . . . have a phase oscillation. Their spatial part doesn’t change – the probability of ”finding the electron at location x” is constant over time – but the complex phase oscillates, and at a rate which is dependent on energy. (”Complex” here meaning ”using imaginary numbers”, not ”complicated”.) Higher energy levels rotate their phase faster than lower ones.

That seemed familiar, but I had a hard time figuring out why. It wasn’t just because I had seen it before, in quantum mechanics and quantum chemistry and quantum computing. Rather, it reminded me of something else, but I had no idea what.

After a couple of weeks of frustration, I finally remembered what it reminded me of: gravitational time dilation in General Relativity. There, clocks that are higher up in a gravitational potential run faster than ones lower down. Now, one of the biggest outstanding problems in modern physics is trying to reconcile General Relativity with Quantum Mechanics. Many people have tried and failed; it’s considered to be a very hard problem. But here we had a case of GR and QM saying similar things. Maybe quantum gravity wasn’t as hard as everyone thought? Maybe this was an easy way in?

So, as an exercise, I decided to see if I could discover the mathematical relationship between those two things. Might they be connected in some way? That would be interesting.

The only real problem was that they are expressed in different terms. One is a phase rotation rate in terms of energy. The other is a time shift in terms of potential. But it’s not too hard to work that through; it only takes high school algebra. One can easily convert the phase into time by using the normal oscillation frequency. And one can easily get energy equivalents for the potential by assuming a ”test particle” with small (but arbitrary) mass m.

In GR the answer is that the rate of time flow is proportional to the total energy (potential energy plus rest mass energy). The time dilation seen for observer b by observer a is just T_b/T_a = E_b/E_a, where E_a and E_b are the total energy of the test particle.

In QM the answer is that the relative rate of phase oscillation (= time flow) is just T_b/T_a = E_b/E_a, where E_a and E_b are the total energy of the electron (including its rest mass).

In other words, charged particles in an electric potential appear to be time-dilated in exactly the same way as massive particles in a gravitational potential. The equations are identical, if you consider the particle’s phase frequency to be its “local clock”.

There followed much thought, crunching of equations, analysis of experiments, and searching the literature. For a few weeks, giddy, I thought I was the first human ever to discover this.

Until I found out I wasn’t.

Reflections

As I meditate here, tranquil,
    by this gently burbling stream,
dappled with patches of afternoon sunlight,
    autumnally serene,
a tiny breeze troubles the placid pool -
    it trembles, and I must respond.
I sense you, my mirror-brother, by
    your equally peaceful pond.
Your empire just as vast as mine,
    your castle just as steep,
its guardians just as many and
    its treasure-hoard as deep.

Pity the poor leaf, swirling by,
    lost in its confusion:
we're beyond that, you and I,
    we have no illusions.

Still, the puzzle of this push and pull,
    this yin-and-yang of ours,
is a mystery I cannot see
    though I stare and stare for hours.
The twisting, flowing dance we dance
    has reached a sorry state.
Our balance is precarious.
    We both just sit and wait.
My slightest motion would send out ripples
    which you would swiftly sense,
so our whirling waltz of force is frozen,
    implacable in defense.

Yet at the heart, where we are one,
    is a single triumphant cry:
both of us willing in an instant to kill -
    in an instant, to die.

Boulder Creek, July 18, 1998
revised December 1998 – January 1999

Is this a study of two contemplative but embattled warlords, or a parable about a particle and antiparticle skittering Leidenfrost-like around the barrier potential that keeps them from annihilating each other in a blinding flash of pure energy? The entire poem is a triple-entendre, with every line having (at least) three meanings. It could just as easily have been entitled Symmetry, or Warlords, or Particle Physics.

Energy = Mass = Frequency ?

Now what we measure when we measure the mass of an electron is the rate that it goes around. The mass of an electron is its energy. Mass and energy are equivalent, as Einstein showed, and energy is equivalent to frequency, as I guess it was de Broglie showed, and so I keep saying mass and energy and frequency interchangeably because I’m so used to that.

– Richard Feynman, The Douglass Robb Memorial Lectures, part 4, University Of Auckland, New Zealand (1979) http://vega.org.uk/video/programme/48

Actually Feynman was slightly off. Einstein of course (in 1905) came up with “E = mc²“, but it was Planck (in 1901) who came up with “E = hν”, where ν is the frequency, in order to explain black body radiation, and later (with Einstein in 1905) the photoelectric effect. Planck’s formula started the chain of dominoes toppling that led through Bohr, Heisenberg, Schrödinger and others to Quantum Mechanics.

Two things that are equal to the same thing are equal to each other, but it took 18 years before de Broglie put the two equations together in his 1923 PhD thesis and got “hν = mc²“. So de Broglie gets the credit for finally connecting all three.

The simplest way of explaining my upcoming experiment is this: I plan to look for changes in the mass of the electron at electric potentials of roughly + and – 1 million volts, conditions for which quantum theory would predict its frequency (and hence mass) is substantially different, but classical electromagnetism predicts no effect whatsoever. Maxwell and de Broglie can’t both be right; they predict very different things, and (at least) one of them has to be very wrong.

This is, sadly, not a direct test of quantum time dilation. Since electrons have infinite lifetimes and don’t decay at all, there is no way to make them decay faster or slower. But the foundation of (my version of) the theory is the de Broglie mass-frequency relation. If the de Broglie relation is wrong, then my path into the theory collapses. Of course, many other things also collapse, including QM. At least I’ll have good company.

On the other hand, if Maxwell is wrong, then other things collapse. Either way, I think we get some new physics. Now, we already know that Maxwell is not completely right, since classical EM does not entirely agree with Quantum Electrodynamics. Maxwell cannot explain entangled light, the photoelectric effect, or indeed even the existence of photons. He cannot explain the hydrogen atom. On the other hand, radios and TVs and cellphones and microwave ovens use the Maxwell equations quite successfully. So the question isn’t whether Maxwell is wrong – of course he is, at least sometimes! – but whether he is MUCH wrong about THIS. Wrong enough for me to measure. I’m looking forward to finding out.