The stage: spacetime and its interval
Special relativity taught us to stop thinking of space and time separately and to work in four-dimensional spacetime. Its key quantity is the invariant interval ds between two nearby events — a 'distance' in spacetime that every observer agrees on, even though they disagree about the separate space and time gaps. In the flat spacetime of special relativity, with Cartesian coordinates and signature $(-,+,+,+)$:
The Minkowski interval. The minus sign in front of time is what distinguishes spacetime from ordinary 4D space.
For a massive body moving slower than light, ds^2<0, and we define the proper time \tau — the time actually ticked by a clock carried along that worldline. It is the interval measured in the body's own frame:
Proper time is the geometric length of a worldline. Remember this — free fall will turn out to maximise it.
The metric tensor: geometry in one object
To describe a curved spacetime we cannot rely on the fixed Minkowski coefficients. Instead we let the coefficients vary from place to place and package them into the metric tensor g_{\mu\nu}. The interval becomes a sum over all coordinate pairs (with the Einstein convention that repeated indices are summed):
The metric is the master object of general relativity: give me g and I can compute every length, angle, time and causal relation.
Curvature you can measure from the inside
How can a being trapped inside a space tell whether it is curved, with no outside vantage point? Gauss answered this for surfaces: curvature is intrinsic, detectable by measurements made entirely within the surface. Draw a large triangle: on a flat plane its angles sum to exactly 180^\circ; on a sphere they sum to more; on a saddle, less. Measure the circumference of a circle of radius r: on a curved surface it is not 2\pi r.
Why time curvature does the heavy lifting
Here is a subtlety worth its weight in gold. Near a mass, the metric component that dominates the motion of slow bodies is the time-time part g_{tt}. In the weak-field limit it takes the form (with \Phi the familiar Newtonian gravitational potential, e.g. \Phi=-GM/r):
The Newtonian potential is buried in the rate at which time flows. Deeper in the well (more negative Phi), clocks run slower.
This single component reproduces all of Newtonian gravity for everyday, slow-moving matter — as Guide 3 will show explicitly. Intuitively, an apple at rest still advances through time; because its clock ticks slightly slower nearer the ground, its straightest path through spacetime bends downward, which we experience as falling. For a fast object like light, the spatial parts of the metric contribute just as much — which is exactly why, in Guide 5, light bends by twice the naive amount.
Working with the metric: raising and lowering indices
The metric also gives us the tool to move between the two flavours of components — contravariant (upper) and covariant (lower). Its matrix inverse g^{\mu\nu} is defined by:
The inverse metric raises indices (V^mu = g^{mu nu} V_nu); the metric itself lowers them. This is the everyday grammar of tensor calculations.
With the metric in hand we can measure the length of any four-vector, time any worldline, and identify the light cones that separate cause from effect. What we cannot yet do is say how a free particle actually moves through this geometry. That is the next guide: geodesics.