Covariance and Contravariance in PhysicsBrian BeckmanMicrsosoft13 Oct 2009

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 conceptsyou think you've got it, then *smack*

something goes backwards Why? Can we fix that?

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)

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

Analyze the Words

"Co- and Contra-" imply dualityThey go together

"Variance" implies movementThey show up when something changes

A Running Example

Imagine a flight plan from Reykjavik to Johannesburg

Imagine two functions:Waypoint as a function of timeFuel as a function of waypoint

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

Time Waypoint

t1 Belfast

t2 LeMans

t3 Perpignon

t4 Bejaia

...

Waypoint

Fuel

Belfast f1

LeMans f2

Perpignon f3

Bejaia f4

...

p t f p

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

…

f p t f p t

df df dp

dt dp dt

df dp dp dt

f p t

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

f fx x

fx

ppx x

f p f p f p x xx x

pf p f x x

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...

p pd f dd f p df

dt dt d dt x x

x x

x

p pp

d ddff f

d dt dt x

x x

x xx

x

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?

p pd df ddf

d dt d dt yx

x y

x y

Intuition by Example

Let Coordinates are numbers relating to

geometry When x is 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

, 1p t a p t a y x

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

1df df dfd

d d d d a y x xx

y x y x

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

p p p p

p

d d d da

dt dt d dt

y x y x

x

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

changesnon-linear, curved, twisted, ...with many kinds of singularitiesthis is where many interesting details

go…

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

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!

p pd f dd f p df

dt dt d dt x x

x x

x

d f p dt f p

p p pf t f t t x xx x x

The Better Notation

is linear approx to at

is linear approx to at

is linear approx to at

pf t x x pfx x t

pf t x x fx p tx

p tx px

p p pf t f t t x xx x x

t

Now Change Coordinatesp p

p p

p p p p

f f

f f

f p f x f f f

x y

y x

x x y y

x X y

y Y x

Y

X

X y y Y x

p p p

p p p p

p p

p p p p

f p t f t f t t

f t f t f t t

f t t

t t t t

y y

y x x

x

y y y

y X y X y X y

x X y

y Y x Y x x

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!

p p pf t f t t y xy x X y

p p pt t t y Y x x