Why the ordinary derivative breaks on a curved space
You reach this final guide of the rung already carrying everything you need. From the earlier guides you have the language of index notation and the Einstein summation convention (a repeated up-down index pair is silently summed), and above all the [[metric-tensor|metric tensor]] g_ij, the little machine that measures lengths and angles on a curved space through the line element ds^2 = g_ij dx^i dx^j. What is missing is calculus. We can measure on a curved space, but we cannot yet differentiate a vector field on it honestly — and that gap is the whole subject of this guide.
Recall the derivative from Volume I: to differentiate, you compare a vector here with a vector a hair's breadth away, subtract, and divide by the tiny step. That subtraction is the trap. On a flat plane with Cartesian axes the basis arrows e_x and e_y point the same way everywhere, so subtracting two nearby vectors is fair. But switch to polar coordinates, or step onto a sphere, and the basis vectors themselves swing around as you move — the radial direction at one point is not the radial direction next door. Subtracting components alone now compares apples to oranges, because you have ignored that the measuring grid rotated underneath you.
Christoffel symbols: the bookkeeping of a turning grid
The fix begins by writing down exactly how much each basis vector turns as you take one step. Differentiate the basis vector e_i in the direction j; the result is again a vector, so it must be a combination of the basis vectors, with coefficients we name the [[christoffel-symbol|Christoffel symbols]] Gamma^k_ij. Read the symbol out loud: 'the amount of the k-th basis arrow you pick up when you nudge the i-th basis arrow in the j-direction'. They are not a tensor — they vanish in Cartesian coordinates and reappear in polar ones for the very same flat plane — which is the giveaway that they encode the GRID's behaviour, not the space's. They are pure bookkeeping for a turning frame.
The beautiful part is that you never have to guess these coefficients: the metric alone determines them. Once you know g_ij at every point, the Christoffel symbols are fixed by a formula built entirely from partial derivatives of the metric, Gamma^k_ij = (1/2) g^km ( dg_mi/dx^j + dg_mj/dx^i - dg_ij/dx^m ), where g^km is the inverse metric. So the metric — the same object that gave you distances and angles — quietly tells you how the coordinate frame twists from point to point. Everything downstream, the derivative, the straightest paths, the curvature, is therefore computable from g_ij and nothing else.
The covariant derivative: an honest derivative at last
Now assemble the cure. To differentiate a vector field V honestly, take the ordinary partial derivative of its components AND add a correction that puts back the turning of the basis. That total is the [[covariant-derivative|covariant derivative]], written nabla_j V^k = dV^k/dx^j + Gamma^k_jm V^m. The first term is the naive change in the numbers; the second term, the Christoffel piece, subtracts off the illusion caused by the grid swinging around. What survives is the genuine change of the vector itself — the part that would be reported identically no matter which coordinates you chose. Unlike the plain partial derivative, the covariant derivative is a true tensor, transforming correctly under any change of coordinates.
THE COVARIANT DERIVATIVE (raise the derivative to a curved space)
nabla_j V^k = dV^k/dx^j + Gamma^k_jm V^m
\---------/ \-----------/
naive change correction:
of the numbers undo the turning of the basis
built from the metric alone:
Gamma^k_ij = (1/2) g^km ( dg_mi/dx^j + dg_mj/dx^i - dg_ij/dx^m )
sanity checks:
flat + Cartesian -> all Gamma = 0 -> nabla = ordinary partial
metric-compatible -> nabla_k g_ij = 0 (the ruler never 'drifts')One property anchors the whole construction: the covariant derivative of the metric is zero, nabla_k g_ij = 0, called metric compatibility. In plain terms, the ruler you measure with never changes length under its own differentiation — your notion of distance is carried along consistently. This single demand, together with the requirement that there be no twisting (symmetry of Gamma in its lower indices), pins down the Christoffel symbols uniquely; that is why the formula above is forced, not chosen. The covariant derivative is the one and only honest way to differentiate that respects the geometry the metric already laid down.
Parallel transport and the straightest possible path
With an honest derivative in hand, two grand ideas fall out at once. The first is [[parallel-transport|parallel transport]]: carrying a vector along a path while keeping it 'as constant as the space allows', meaning its covariant derivative along the path is zero. You are not freezing the components — you are letting them change by exactly the Christoffel amount so the real arrow stays steady. The startling fact is that the answer depends on the route: parallel-transport a vector around a closed loop on a sphere and it comes back rotated by an angle equal to the area enclosed. That path-dependence is curvature announcing itself, and it has no analogue on the flat plane, where a loop always brings a vector back unchanged.
- Pick a path through the curved space and a starting vector V sitting at its first point.
- Step forward a tiny bit. The basis arrows have rotated slightly, so the components of V must change by exactly minus the Christoffel correction to keep the real arrow pointing the same physical way.
- Repeat step after step; this is just solving the equation 'covariant derivative of V along the path = 0' bit by bit.
- Now special-case it: if the vector you transport IS the path's own velocity, demanding it stay parallel to itself gives the geodesic equation — the straightest path.
That last step delivers the second grand idea: the [[calc-geodesic|geodesic]], the curved-space version of a straight line. A straight line is the path that never turns; on a curved space 'never turns' means parallel-transports its own velocity, giving the geodesic equation d^2x^k/dt^2 + Gamma^k_ij (dx^i/dt)(dx^j/dt) = 0. Set the Christoffel symbols to zero (flat, Cartesian) and it reads d^2x^k/dt^2 = 0, whose solutions are ordinary straight lines at constant speed — Newton's first law. Geodesics are also the shortest paths, which you can derive as a Euler-Lagrange problem of minimizing the length integral of ds; on a sphere they are the great circles, which is exactly why long-haul flights arc poleward.
Curvature, and how it became gravity
We can finally name the deepest object on the rung. On a flat space, taking the covariant derivative twice in different orders gives the same answer — order does not matter. On a curved space it does, and the gap between 'first along i then j' and 'first along j then i' is measured by the [[riemann-curvature-tensor|Riemann curvature tensor]] R^l_kij. Built from the Christoffel symbols and their derivatives, it is the honest curvature: it stays zero in every coordinate system if and only if the space is truly flat, no matter how twisted the coordinates look. It is precisely the rotation a vector picks up after parallel transport around an infinitesimal loop, packaged as a tensor.
This is the whole machine that becomes general relativity. Einstein's leap was to treat spacetime itself as a curved space with its own metric, where the geodesics are the paths that freely falling objects follow. A planet orbiting the Sun is not being tugged by a force at all in this picture — it is coasting along the straightest available path through a spacetime that the Sun's mass has curved. Gravity is not a force pulling things off straight lines; it is the bending of the straight lines themselves. Einstein's field equations then say, in one tensor line, how much matter and energy curve spacetime: the curvature on the left equals the matter-energy on the right.