Particle Physics

Mr. Einstein and my GPS

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I promised to continue one of my previous posts and explain how Einstein’s theories of 1905 and 1915 together affect our GPS systems. If we hadn’t discovered relativity (special and general) by now, we’d have certainly discovered it by the odd behaviour of our clocks on the surface of the earth and on an orbiting satellite.

The previous post ended by demonstrating that the time interval between successive ticks of a clock at the earth’s surface \Delta t_R and a clock ticking infinitely far away from all masses \Delta t_{\infty} are related by the formula

\Delta t_{R} =  \Delta t_{\infty} (1 + \frac{ \Phi(R)}{c^2})

The gravitational potential \Phi(R)=-\frac{G M_E}{R} is a {\bf {negative}} number for all R. This means that the time intervals measured by the clock at the earth’s surface is {\bf {shorter}} than the time interval measured far away from the earth. If you saw the movie “Interstellar“, you will hopefully remember that a year passed on Miller’s planet (the one with the huge tidal waves) while 23 years passed on the Earth, since Miller’s planet was close to the giant Black Hole Gargantua. So time appears to slow down on the surface of the Earth compared to a clock placed far away.

Time for some computations. The mass of the earth  is 5.97 \times 10^{24} \: kg, Earth’s radius is R = 6370 \: km \: = 6.37\times 10^6 \: meters and the MKS units for G = 6.67 \times 10^{-11} \: MKS \: units. In addition, the speed of light c = 3 \times 10^8 \frac {m}{s}. If \Delta t_{\infty} = 1 \: sec, the clock on an orbiting satellite (assumed to be really far away from the earth) measures one second, the clock at the surface measures

\Delta t_R = (1 \: sec) \times (1 - \frac {(6.67 \times 10^{-11}) \:  \times \:  (5.97 \times 10^{-24} )}{(6.37 \times 10^6 )\: (3 \times 10^8 )^2})

this can be simplified to  0.69 \: nanoseconds less than 1 \: sec. In a day, which is  (24 \times 3600) \: secs, this is 70 \times 10^-6 = 60 \: \mu \: seconds (microseconds are a millionth of a second).

In reality, as will be explained below, the GPS satellites are operating at roughly 22,000 \: km above the earth’s surface, so what’s relevant is the {\bf {difference}} in the gravitational potential at 28,370 \: km and 6,370 \: km from the earth’s center. That modifies the difference in clock rates to 53 \: nanoseconds per second, or 46 \: microseconds in a day.

How does GPS work? The US (other countries too – Russia, the EU, China, India) launched several satellites into a distant orbit 20,000 \: - 25,000 \: km above the earth’s surface. Most of the orbits are designed to allow different satellites to cover the earth’s surface at various points of time. A few of the systems (in particular, India’s) have satellites placed in a Geo-Stationary orbit, so they rotate around the earth with the earth – they are always above a certain point on the earth’s surface. The key is that they possess rather accurate and synchronized atomic clocks and send the time signals, along with the satellite position and ID to GPS receivers.

If you think about how to locate someone on the earth, if I told you I was 10 miles from the Empire State Building in Manhattan, you wouldn’t know where I was. Then, if I told you that I was 5 miles from the Chrysler building (also in Manhattan), you would be better off, but you still wouldn’t know how high I was. If I receive a third coordinate (distance from yet another landmark), I’d be set.  So we need distances from three well-known locations in order to locate ourselves on the Earth’s surface.

The GPS receiver on your dashboard receives signals from three GPS satellites. It knows how far they are, because it knows when the signals were emitted, as well as what the time at your location is.  Since these signals travel at the speed of light (and this is sometimes a problem if you have atmospheric interference), the receiver can compute how far away the satellites are. Since it has distances to three “landmarks”, it can be programmed to compute its own location.

Of course, if its clock was constantly running slower than the satellite clocks, it would constantly overestimate the distance to these satellites, for it would think the signals were emitted \: earlier than they actually were. This would screw up the location calculation, to the distance travelled by light in 0.53 \: nanoseconds, which is 0.16 meters. Over a day, this would become 14 kilometers. You could well be in a different city!

There’s another effect – that of time dilation. To explain this, there is no better than the below thought experiment, that I think I first heard of from George Gamow’s book. As with {\bf {ALL}} arguments in special and general relativity, the only things observers can agree on is the speed of light and (hence) the order of causally related events. That’s what we use in the below.

There’s an observer standing in a much–abused rail carriage. The rail carriage is travelling to the right, at a high speed V. The observer has a rather cool contraption / clock. It is made with a laser, that emits photons and a mirror, that reflects them. The laser emits photons from the bottom of the carriage towards the ceiling, where the mirror is mounted. The mirror reflects the photon back to the floor of the car, where it is received by a photo-detector (yet another thing that Einstein first explained!).

Light Clock On Train

The time taken for this up- and down- journey (the emitter and mirror are separated by a length L) is

\Delta t' = \frac{2 L}{c}

That’s what the observer on the train measures the time interval to be. What does an observer on the track, outside the train, see?

Light Clock Seen from Outside Train

She sees the light traverse the path down in blue above. However, she also sees the light traveling at the same (numerical) speed, so she decides that the time between emission and reception of the photon is found using Pythagoras’ theorem

L^2 = (c \frac{\Delta t}{2})^2 - (V \frac {\Delta t}{2})^2

\rightarrow  \Delta t = \frac {2 L}{c} \frac{1}{\sqrt{1 - \frac{V^2}{c^2}}}

So, the time interval between the same two events is computed to be larger on the stationary observer’s clock, than on the moving observer’s clock. The relationship is

\Delta t = \frac {\Delta t'}{ \sqrt{1 - \frac{V^2}{c^2}} }

How about that old chestnut – well, isn’t the observer on the track moving relative to the observer on the train? How come you can’t reverse this argument?

The answer is – who’s going to have to turn the train around and sheepishly come back after this silly experiment runs its course? Well! The point is that one of these observers has to actively come back in order to compare clocks. Relativity just observes that you cannot make statements about {\bf {absolute}} motion. You certainly have to accept relative motion and in particular, how observers have to compare clocks at the same point in space.

From the above, 1 second on the moving clock would correspond to \frac {1}{ \sqrt{1 - \frac{V^2}{c^2}} } seconds on the clock by the tracks. A satellite at a distance D from the center of the earth has an orbital speed of \sqrt {\frac {G M_E}{D} } , which for an orbit 22,000 km above the earth’s surface, which is 28,370 \: km from the earth’s center, would be roughly

\sqrt { \frac {(6.67 \times 10^{-11} (5.97 \times 10^{-24})}{28370 \times 10^3} }\equiv  3700 \: \frac{meters}{sec}

which means that 1 second on the moving clock would correspond to 1 \: sec + 0.078 \: nanoseconds on the clock by the tracks. Over a day, this would correspond to a drift of 6 \: microseconds, in the {\bf {opposite}} direction to the above calculation for gravitational slowing.

Net result – the satellite clocks run faster by 40 microseconds in a day. They need to be continually adjusted to bring them in sync with earth-based clocks.

So, that’s three ways in which Mr. Einstein matters to you EVERY day!

 

Special Relativity; Or how I learned to relax and love the Anti-Particle

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The Special Theory of Relativity, which is the name for the set of ideas that Einstein proposed in 1905 in a paper titled “On the Electrodynamics of moving bodies”, starts with the premise that the Laws of Physics are the same for all observers that are traveling at uniform speeds relative to each other. One of the Laws of Physics includes a special velocity – Maxwell’s equations for electromagnetism include a special speed c, which is the same for all observers. This leads to some spectacular consequences. One of them is called the “Relativity of Simultaneity”. Let’s discuss this with the help of the picture below.

CKansen-ToUse

Babu is sitting in a railway carriage, manufactured by the famous C-Kansen company, that travels at speeds close to that of light. Babu’s sitting exactly in the middle of the carriage and for reasons best known to himself (I guess the pantry car was closed and he was bored), decides to shoot a laser beam simultaneously at either end of the carriage from his position. There are detectors/mirrors that detect the light at the two ends of the carriage. As far as he is concerned, light travels at 3 \times 10^5 \frac {km}{sec} and he will challenge anyone who says otherwise to a gunfight – note that he is wearing a cowboy hat and probably practices open carry.

Since the detectors at the end of the carriage are equidistant from him, he is going to be sure to find the laser beams hit the detectors simultaneously, from his point of view.

Now, consider the situation from the point of view of Alisha, standing outside the train, near the tracks, but safely away from Babu and his openly carried munitions. She sees that the train is speeding away to the left, so clearly since {\bf she} thinks light also travels at 3 \times 10^5 \frac {km}{sec}, she would say that the light hit the {\bf right} detector first before the {\bf left} detector. She doesn’t {\underline {at \: all \: think}} that the light hit the two detectors simultaneously. If you asked her to explain, she’d say that the right detector is speeding towards the light, while the left detector is speeding away from the light, which is why the light strikes them at different times.

Wait – it is worse. If you had a third observer, Emmy, who is skiing to the {\bf {left}} at an even higher speed than the C-Kansen (some of these skiers are crazy), she thinks the C-Kansen train is going off to the right (think about it), not able to keep up with her. As far as {\underline {\bf {she}}} is concerned, the laser beam hit the {\bf {left}} detector before the other beam hit the {\bf {right}} detector.

What are we finding? The Events in question are – “Light hits Left Detector” and “Light hits Right Detector”. Babu claims the two events are simultaneous. Alisha claims the second happened earlier. Emmy is insistent that the first happened earlier. Who is right?

They are ALL correct, in their own reference frames. Events that appear simultaneous in one reference frame can appear to occur the one before the other in a different frame, or indeed the one after the other, in another frame. This is called the Relativity of Simultaneity. Basically, this means that you cannot attribute one of these events to have {\bf {caused}} the other, since their order can be changed. Events that are separated in this fashion are called “space-like separated”.

Now, on to the topic of this post. In the physics of quantum field theory, particles interact with each other by exchanging other particles, called gauge bosons. This interaction is depicted, in very simplified fashion so we can calculate things like the effective force between the particles, in a sequence of diagrams called Feynman diagrams. Here’s a diagram that depicts the simplest possible interaction between two electrons

MollerScattering

Time goes from the bottom to the top, the electrons approach each other, exchange a photon, then scoot off in different directions.

This is the simplest diagram, though and to get the exact numerical results for such scattering, you have to add higher orders of this diagram, as shown below

MoreMoller

When you study such processes, you have to perform mathematical integrals – all you know is that you sent in some particles from far away into your experimental set-up, something happened and some particles emerged from inside. Since you don’t know where and when the interaction occurred (where a particle was emitted or picked up, as at the vertexes in the above diagrams), you have to sum over all possible places and times that the interaction {\bf {could}} have occurred.

Now comes the strange bit. Look at what might happen when you sum over all possible paths for a collision between an electron and a photon.

e+e--Fig1

In the above diagram, the exchange was simultaneous.

In the next one, the electron emitted a photon, then went on to absorb a photon.

e+e--Fig2

and then comes the strange bit –

e+e--Fig3

 

Here the electron emitted a photon, then went backwards in time, absorbed a photon, then went its way.

When we sum over all possible event times and locations, this is really what the integrals in quantum field theory instruct us to do!

Really, should we allow ourselves to count processes where  two events occur simultaneously, which means we would then have to allow for them to happen in reverse order, as in the third diagram? What’s going on? This has to be wrong! And what’s an electron going backwards in time anyway? Have we ever seen such a thing?

Could we simply ban such processes? So, we would only sum over positions and times where the intermediate particles had enough time to go from one place to another,

There’s a problem with this. Notice the individual vertexes where an electron comes in, emits (or absorbs) a photon, then moves on. If this were a “real” process, it wouldn’t be allowed. It violates the principle of energy-momentum conservation. A simple way to understand this is to ask, could a stationary electron suddenly emit a photon and shoot off in a direction opposite to the photon, It looks deceptively possible! The photon would have, say, energy E and momentum p = E/c. This means that the electron would also have momentum E/c, in the opposite direction (for conservation) but then its energy would have to be \sqrt{E^2+m^2 c^4} from the relativistic formula. This is higher than the energy m c^2 of the initial electron : A+ \sqrt{A^2+m^2} is bigger than m! Not allowed !

We are stuck. We have to assume that energy-momentum conservation is violated in the intermediate state – in all possible ways. But then, all hell breaks loose – in relativity, the speed of a particle v is related to its momentum p and its energy E by v = \frac {p}{E} – since p and E can be {\underline {anything}}, the intermediate electron could, for instance, travel faster than light. If so, in the appropriate reference frame, it would be absorbed before it was created. If you can travel faster than light, you can travel backwards in time (read this post in Matt Buckley’s blog for a neat explanation).

If the electron were uncharged, we would probably be hard-pressed to notice. But the electron is charged. This means if we had the following sequence of events,

– the world has -1 net charge

– electron emits a photon and travels forward in time

– electron absorbs a photon and goes on.

This sequence doesn’t appear to change the net charge in the universe.

But consider the following sequence of events

– the world has -1 net charge

– the electron emits a photon and travels backwards in time

– the electron absorbs a photon in the past and then starts going forwards in time

Now, at some intermediate time, the universe seems to have developed two extra negative charges.

This can’t happen – we’d notice! Extra charges tend to cause trouble, as you’d realize if you ever received an electric shock.

The only way to solve this is to postulate that an electron moving backward in time has a positive charge. Then the net charge added for all time slices is always -1.

ergo, we have antiparticles. We need to introduce the concept to marry relativity with quantum field theory.

There is a way out of this morass if we insist that all interactions occur at the same point in space and that we never have to deal with “virtual” particles that violate momentum-energy conservation at intermediate times. This doesn’t work because of something that Ken Wilson discovered in the late ’70s, called the renormalization group – the results of our insistence would be we would disagree with experiment – the effects would be too weak.

For quantum-field-theory students, this is basically saying that the expansion of the electron’s field operator into its components can’t simply be

\Psi(\vec x, t) = \sum\limits_{spin \: s} \int \frac {d \vec k}{(2 \pi)^3} \frac{1}{\sqrt{2 E_k}} b_s(\vec k) e^{- i E_k t + i {\vec k}.{\vec x}}

but has to be

\Psi(\vec x, t) = \sum\limits_{spin \: s} \int \frac {d \vec k}{(2 \pi)^3} \frac{1}{\sqrt{2 E_k}} \left( b_s(\vec k) e^{- i E_k t + i {\vec k}.{\vec x}}  +d^{\dagger}_s(\vec k) e^{+ i E_k t - i {\vec k}.{\vec x}} \right)

including particles being destroyed on par with anti-particles being created and vice versa.

The next post in this sequence will discuss another interesting principle that governs particle interactions – C-P-T.

Quantum field theory is an over-arching theory of fundamental interactions. One bedrock of the theory is something called C-P-T invariance.  This means that if you take any physical situation involving any bunch of particles, then do the following

  • make time go backwards
  • parity-reverse space (so in three-dimensions, go into a mirror world, where you and everything else is opposite-handed)
  • change all particles into anti-particles (with the opposite charge)

then you will get a process which could (and should) happen in your own world. As far as we know this is always right, in general and it has been proven under a variety of assumptions. A violation of the C-P-T theorem in the universe would create quite a stir. I’ll discuss that in a future post.

Addendum: After this article was published, I got a message from someone I respect a huge amount that there is an interesting issue here. When we take a non-relativistic limit of the relativistic field theory, where do the anti-particles vanish off to? This is a question that is I am going to try and write about in a bit!