Introduction to General Relativity

Gravity as spacetime curvature and its measurable effects. Theory, formulas, worked examples and interactive graphs.

Conceptual context

General relativity changes the meaning of gravity. Instead of treating it only as a force, it describes how $M$ shapes spacetime geometry and how that geometry changes clock rates. This leaf uses the clean exterior Schwarzschild case: a spherical, stationary, nonrotating source. The central question is how $\alpha$ varies when a stationary clock is placed at different $r$ values relative to $r_s$. The graph is not a path through space. It is a comparison between local $\Delta \tau$ and distant $\Delta t$.

🟢 Essential level

A stationary clock near a spherical $M$ does not tick at the same rate as a clock very far away. The clock is not damaged; rather, the spacetime geometry around $M$ changes the relation between local and distant time. $\alpha$ tells what fraction of distant $\Delta t$ appears as local $\Delta \tau$. Far from the source, the factor approaches one and both clock rates nearly agree. When $r$ approaches $r_s$, the factor becomes much smaller and the local clock accumulates less time. Therefore the graph rises as we move outward: larger radial distance means weaker curvature for the stationary clock and a time rate closer to the distant reference.

This is a comparison of clocks at fixed positions, not a story about a clock flying through space or changing its mechanism. It is like two identical metronomes, one at the foot of a mountain and one at its summit: neither is broken, but the gravitational "altitude" each one lives at makes them tick at slightly different rates.

🔵 Formal level

Basic formulas

The model uses the exterior Schwarzschild geometry. The first relation defines the radial scale of the source, $r_s$, from $M$, $G$, and $c$.

Schwarzschild radius

The second relation compares the stationary clock position $r$ with that scale. It returns $\alpha$, a dimensionless quantity. The expression is physically read in the exterior stationary domain, so $r$ must be greater than $r_s$.

Gravitational time dilation factor in Schwarzschild spacetime

Real student questions

Does the clock stop working near $M$? No. It records its own $\Delta \tau$ normally; the difference appears when compared with distant $\Delta t$.

Is $r_s$ always the object's physical radius? No. It is a gravitational scale. A planet normally has a material radius far larger than its $r_s$.

Why use r/$r_s$ on the graph? It lets different spherical sources share the same curve shape while keeping physical scale separate.

Can this describe a rotating black hole? Not directly; that requires an extended model.

Final synthesis

In the exterior Schwarzschild model, gravity changes stationary clock rates. $\alpha$ increases with $r$ and approaches one far from the spherical source.