ISEGORIA / MATH ENCYCLOPEDIA
General relativity: curvature, clocks, and light
Use a labeled embedding diagram, gravitational time dilation, and weak-field light deflection.
Before you begin: Special relativity and differential geometry
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. An embedding diagram
Outside a non-rotating mass, the geometry of the equatorial plane at one moment can be drawn as a curved surface in an imagined extra dimension. Change the Schwarzschild radius and drag the gold ring: the length along the profile from the horizon to r is the proper distance a chain of rulers would measure, and it is always longer than r − rₛ, because the radius r is defined by the circumference 2πr.
Worked example. At r = 2rₛ the proper distance from the horizon is (√2 + ln(1 + √2)) rₛ ≈ 2.30 rₛ, more than twice the coordinate difference rₛ.
Watch out. The extra vertical axis is only a device for drawing the curved space of one time slice. The surface does not show the curvature of time, which is what makes things fall, and the region inside the horizon is not drawn.
What is the extra vertical axis?
It is a visualization coordinate used to display intrinsic spatial geometry.
2. Gravitational time dilation
Hold a clock at rest at radius r outside a black hole and compare it with a clock far away. Drag the blue clock: for every tick of the far clock it advances only √(1 − rₛ/r), and light it sends outward arrives redshifted by the inverse factor. The dashed curve is the weak-field estimate 1 + Φ/c², which is good far out and fails near the horizon.
Worked example. At r = 2rₛ the factor is √(1/2) ≈ 0.707: the near clock loses about 17.6 minutes for every hour of the far clock.
Watch out. The formula is for a clock held static outside a non-rotating spherical mass, which needs a rocket or a support; a clock falling or orbiting also has a speed-dependent factor.
Which clock accumulates less proper time?
The clock deeper in the gravitational field.
3. Light bending near a black hole
Light passing a mass is bent. Here each ray is traced by integrating the exact orbit equation for light in the Schwarzschild geometry, and its deflection is compared with the weak-field formula 2rₛ/b. Drag the impact parameter b: far out the two agree, near the photon sphere the true angle grows without bound, and below b = (3√3/2) rₛ the light is captured.
Worked example. For the Sun, rₛ ≈ 2.95 km and b ≈ 6.96 × 10⁵ km for a grazing ray, so α ≈ 8.5 × 10⁻⁶ rad ≈ 1.75″, the value confirmed at the 1919 eclipse.
Watch out. The weak-field formula is the first term of an expansion in rₛ/b. At b = 5rₛ it already underestimates the angle by about a third, and near the photon sphere it fails completely.
What reduces the deflection?
Increasing the impact parameter b.