Schwarzschild Radius, Mass and Normalized Surface Gravity

Formula and method

Using the retained approximate solar-mass scale of 1.989 ×10³⁰ kg gives a Schwarzschild radius of about 2.954 km. The normalized surface gravity is about 1.52 ×10¹³ m/s².

r_s = 2 G M/c²; normalized surface gravity κ = c⁴/(4 G M) = c²/(2 r_s).

Worked example

Enter these known values and leave the other values blank.

Assumptions and limitations

• This is the ideal Schwarzschild geometry: nonrotating, uncharged and asymptotically flat. Mass, horizon radius and normalized surface gravity must be strictly positive. Zero mass does not describe a black hole and makes the inverse-gravity relationship singular.
• The horizon radius is an areal radius: the horizon sphere has area 4π r_s². It is not a solid surface, a measurement of a singularity’s size or an observer’s proper radial distance.
• The gravity control is the Schwarzschild surface-gravity parameter κ with time normalized at infinity. The captured expression GM/r_s² equals this parameter, but it is not the locally measured acceleration needed to hover at the horizon. That local static-observer acceleration grows without bound as the horizon is approached.
• The two retained relationships use exact c = 299792458 m/s and the 2022 CODATA measured central G = 6.67430 ×10⁻¹¹ m³/(kg·s²). G has standard uncertainty 0.00015 ×10⁻¹¹; displayed numerical precision does not eliminate input or constant uncertainty.
• Earth and Sun mass choices retain approximate scales 5.9722 ×10²⁴ kg and 1.989 ×10³⁰ kg. They are disclosed input conversions, not exact defining constants or current measurements. Enter kilograms for a specific physical mass estimate.
• The solar-radius length choice is the IAU nominal conversion 6.957 ×10⁸ m, not an exact measurement of the actual Sun. One astronomical unit is 149597870700 m; one parsec is (648000/π) astronomical units. A light-year uses exact c and a Julian year of 365.25 days of 86400 seconds.
• The g unit on the gravity control means conventional standard acceleration 9.80665 m/s², not grams. Selecting g changes the unit of κ; it does not change the black-hole model.
• Any of the three positive quantities can drive the retained inverse graph. Locking inconsistent inputs reports a conflict rather than silently treating them as a valid black hole. This model does not predict radiation, evolution, spin, charge or a freely falling observer’s acceleration. This is a classical geometry model. Near or below the Planck mass (about2.18 ×10⁻⁸ kg), a radius from this equation is a formal extrapolation rather than an established description of a physical black hole.

Common questions

References

• Cambridge DAMTP: Schwarzschild surface gravity and normalization
• NIST 2022 CODATA: measured G and exact c
• IAU 2015 B2: astronomical unit and exact parsec convention
• IAU 2015 B3: nominal solar radius
• NIST SP811: time and standard-gravity unit conventions
• Calculation definition and unit reference

Preserve the captured two equations and four inverse branches with current central G and exact c; require strictly positive mass, radius and normalized surface gravity to exclude the singular zero-mass state. Correct rounded light-year and parsec factors to their declared Julian-year and IAU definitions, and label the retained solar radius as a nominal conversion.

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