Covariance and Contravariance in Physics

1. Covariance and Contravariance in Physics Brian Beckman Micrsosoft 13 Oct 2009

2. Covariance and Contravariance • Show up in math, physics, and programming • Different ideas with the same name? • Or facets of one bigger idea? • What's the common thread? • Seem to be slippery concepts • you think you've got it, then *smack* something goes backwards • Why? Can we fix that?

3. Start with the "Devil I Know" • Explore these concepts in physics context • Later, tie them into similar concepts in programming (and maybe other math areas)

4. Ok, What are They in Physics? • They arise from application of "Differential Geometry on Manifolds" • Foundation for General Relativity • GPS, Astrometry, Cosmology, Black holes, ... • Most other areas of physics have been Geometricized • Mechanics, Electrodynamics, Quantum physics, Statistical physics, ... • Looks like String Theory or Loop Quantum Gravity will "seal the deal" • We think all of physics will be Geometricized

5. Analyze the Words • "Co- and Contra-" imply duality • They go together • "Variance" implies movement • They show up when something changes

6. A Running Example • Imagine a flight plan from Reykjavik to Johannesburg • Imagine two functions: • Waypoint as a function of time • Fuel as a function of waypoint

7. Two Functions Call this FLIGHT PATH Call this FUEL • Give it a real-number time t, it gives you a location or point • Give it a point, it gives you a real-number fuel-spent value

8. Composition: fuel over time • – give it a time, you get a fuel-spent • What's the fuel burn-rate? • In 19th-century notation, plus “chain rule” • Great, but what are and ? • Can’t compute them without coordinates …

9. Relate fuel & path to coordinates • Let x be a coordinate function, that gives to every point p on the globe a lat, lon, alt • Define a new function that delivers fuel-spent as a function of coordinates x • Use associative law • Rename (because we will never again separate x from p • Our new fuel-over-time function

10. Now compute fuel burn-rate • Still sticking with "picturesque" 19th-century notation: • You may recognize this as • The burn rate is a gradient times a velocity • The notation is broken, but before fixing it...

11. Here's the Big Idea • The answer CAN'T depend on the choice of coordinate system • The anser MUST be invariant w.r.t. changes in coordinates • We can get this if one of the two factors is covariant and the other is contravariant w.r.t. coordinate change, but which is which?

12. Intuition by Example • Let • Coordinates are numbers relating to geometry • When xis 1, y is a, bigger than x • y is a finer-grained coordinate system • It takes more y's than x's to get from one point to another

13. Gradients are Covariant • The change in f for a unit change in y must therefore be smaller than the change in f for a unit change in x • Check this with the good-ol' chain rule again • When a>1, df/dy is smaller than df/dx • when the coordinate spacing is smaller, the changes in f are smaller • The gradient co-varies with the coordinate spacing

14. Velocities are Contravariant • Chain rule again • When the coordinate spacing gets smaller, the velocity vector must cover more coordinates to represent the same velocity, so its numbers get bigger • The velocity varies contra to (against) the coordinate spacing

15. Intuition versus Precision • That demonstration gives the intuition and the mnemonic • The chain rule gives the precise answer • In any number of dimensions • For any differentiable coordinate changes • non-linear, curved, twisted, ... • with many kinds of singularities • this is where many interesting details go…

16. The Notation is Broken • What is a derivative? • It's the best linear approximation to a function at a certain point • Linear approximation means you give it a change in inputs to the original function, it gives the approximate change in the output of the original function • The derivative is thus a function from points to linear approximations • The derivative operator converts a function of points to a function of points

17. Notation is Broken (cont...) • With all that in mind, what could mean? • This must be a function of time t that produces a linear approximation to • Let's write it like this • as an equality on linear approximations!

18. The Better Notation • is linear approx to at • is linear approx to at • is linear approx to at

19. Now Change Coordinates

20. And Here They Are • Here's the covariant buddy, the gradient • Here's the contravariant buddy, the velocity • This is just the beginning... • But the end for now!