## Schwarzschild metric |

In **Schwarzschild metric** (also known as the **Schwarzschild vacuum** or **Schwarzschild solution**) is the solution to the

According to **Schwarzschild black hole** or **static black hole** is a

The Schwarzschild black hole is characterized by a surrounding spherical boundary, called the

- the schwarzschild metric
- history
- singularities and black holes
- alternative coordinates
- flamm's paraboloid
- orbital motion
- symmetries
- curvatures
- see also
- notes
- references

In

where

- when
*dτ*^{2}is positive, τ is theproper time (time measured by a clock moving along the sameworld line with thetest particle ), - c is the
speed of light , - t is the time coordinate (measured by a stationary clock located infinitely far from the massive body),
- r is the radial coordinate (measured as the circumference, divided by 2π, of a sphere centered around the massive body),
- θ is the
colatitude (angle from north, in units ofradians ), - φ is the
longitude (also in radians), and *r*_{s}is theSchwarzschild radius of the massive body, ascale factor which is related to its mass M by*r*_{s}= 2*GM*/*c*^{2}, where G is thegravitational constant .^{[1]}

This solution is anagolous to a classical Newtonian theory of gravity that corresponds to the gravitational field around a point particle.^{[2]}

The radial coordinate turns out to have physical significance as the "proper distance between two events that occur simultaneously relative to the radially moving geodesic clocks, the two events lying on the same radial coordinate line".^{[3]}

In practice, the ratio *r*_{s}/*r* is almost always extremely small. For example, the Schwarzschild radius *r*_{s} of the Earth is roughly , while the Sun, which is 8.9 mm×10^{5} times as massive 3.3^{[4]} has a Schwarzschild radius of approximately 3.0 km.
Even at the surface of the Earth, the corrections to Newtonian gravity are only one part in a billion. The ratio becomes large only in relatively close proximity to ^{[citation needed]}

The Schwarzschild metric is a solution of *outside* the gravitating body. That is, for a spherical body of radius R the solution is valid for *r* > *R*. To describe the gravitational field both inside and outside the gravitating body the Schwarzschild solution must be matched with some suitable interior solution at *r* = *R*,^{[5]} such as the

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