What Is the Vaidya Metric?

The Vaidya Metric is a solution to Einstein’s equations that describes a spacetime with a radiating body, specifically:

✔ a spherically symmetric mass

✔ that gains or loses mass

✔ by emitting or absorbing null radiation (light-like radiation)

What Is the Vaidya Metric?

This metric was introduced by P. C. Vaidya, an Indian physicist, in 1951.

It is sometimes called the radiating Schwarzschild solution, because it is similar to the Schwarzschild metric but allows mass to change over time.

Why Is the Vaidya Metric Important?

Most celestial objects (stars, black holes) radiate energy — through electromagnetic radiation, neutrinos, particle winds, or Hawking radiation.

The Schwarzschild metric cannot describe such bodies because it assumes the mass is constant.

The Vaidya metric fixes this by allowing:

✔ time-dependent mass ( M(v) )

✔ radiation moving at the speed of light

✔ dynamic horizons

This makes it essential for studying:

• Black hole evaporation
• Black hole growth
• Stellar collapse with radiation
• Accretion phenomena
• Astrophysical explosions
• GR shockwaves and light shells

The Standard Form of the Vaidya Metric

In advanced null coordinates (ingoing radiation):

Where:

• ( v ) = advanced time (suitable for incoming radiation)
• ( M(v) ) = mass depending on time
• Light rays follow null trajectories, hence the name “null dust metric”

There is also a retarded time version for outgoing radiation.

Relation to Black Holes

The Vaidya metric helps describe:

✔ Black Hole Accretion (mass increases)

Matter falling in adds mass → horizon grows.

✔ Hawking Radiation (mass decreases)

Mass slowly decreases → horizon shrinks.
Theoretical models of evaporating black holes must use a Vaidya-like metric.

✔ Dynamic Horizons

Unlike Schwarzschild’s fixed event horizon, Vaidya black holes have:

• evolving apparent horizons
• evolving trapped surfaces
• quasi-local horizons

Where the Vaidya Metric Is Used

1. Supernova Modeling
   During collapse, stars emit intense neutrino radiation → Vaidya describes outer spacetime.
2. Gamma-Ray Bursts
   Radiation shells moving at light speed.
3. Black Hole Growth
   Accretion disks feeding mass into black holes.
4. Hawking Radiation Models
   Quantum radiation emitted outward.
5. Gravitational Collapse
   Used in Oppenheimer–Snyder extensions.
6. Shockwave and Light Shell Models
   e.g., modeling sudden bursts of radiation.

Difference Between Vaidya and Schwarzschild

Feature	Schwarzschild	Vaidya
Mass	Constant	Varies with time (M(v))
Radiation	No	Yes (null radiation)
Horizon	Static	Dynamic, moving
Energy Source	Vacuum	Null dust

The Vaidya metric is more realistic for astrophysical systems.

Intuition (Simple Explanation)

Imagine a star that keeps shining and losing energy:
It becomes slightly lighter every moment.

Spacetime around it changes dynamically, not statically.

This “dynamic spacetime with radiation” is exactly what the Vaidya metric describes.

Final Thoughts

The Vaidya metric is one of the most elegant solutions in General Relativity — simple in form, but powerful in meaning. It bridges the gap between idealized static objects and realistic astrophysical bodies that radiate, shine, collapse, and evolve.