Opening your learning space
Starting the app …
Opening your learning space
Starting the app …
The Schwarzschild radius is the event horizon of an idealized, uncharged, nonrotating, spherically symmetric mass.
Free to browse · flashcards or a multiple-choice quiz · import from Essential
Formula
rs in m; G in m3/(kg·s2); M in kg; c in m/s
| Symbol | Meaning | Unit |
|---|---|---|
| Event-horizon radius in the Schwarzschild model | m | |
| Gravitational constant | m3/(kg·s2) | |
| Mass of the compact object | kg | |
| Speed of light in vacuum | m/s |
r_s=\frac{2GM}{c^2}The Einstein field-equation solution for a spherical, nonrotating, uncharged mass has horizon rs=2GM/c2. A rotating object requires Kerr geometry, so this is not a complete horizon description. The radius is not a material surface. Knowing a mass and radius alone does not make an object a black hole; matter must be concentrated inside the horizon.
The Einstein field-equation solution for a spherical, nonrotating, uncharged mass has horizon rs=2GM/c2. A rotating object requires Kerr geometry, so this is not a complete horizon description. The radius is not a material surface. Knowing a mass and radius alone does not make an object a black hole; matter must be concentrated inside the horizon.
The relation rs=\frac{2GM}{c^2} connects the quantities defined in the variable list. Its form follows from the underlying astronomical model; it is not a universal rule outside the stated assumptions.
Defining relation
This is the documented defining relation. Rearrange it only when the requested quantity can be isolated unambiguously.
Check the worked example for Schwarzschild radius of a nonrotating black hole: For M equal to one solar mass, rs≈2·6.674·10−11·1.989·1030/(2.998·108)2≈2.95 km.
Substitute quantities in the documented order, keep units consistent and compare the result with the check stated in the example.
Applying an approximate relation to real measurements without checking its model range.
Check the measurement definition, assumptions and units before interpreting the result physically.
Formula cluster
Estimate the event-horizon scale of compact astrophysical objects.
For M equal to one solar mass, rs≈2·6.674·10−11·1.989·1030/(2.998·108)2≈2.95 km.
Estimate the event-horizon scale of compact astrophysical objects.
Here is how to work through a typical Schwarzschild radius of a nonrotating black hole (rs = 2GM/c²) task step by step:
Task
Check the worked example for Schwarzschild radius of a nonrotating black hole: For M equal to one solar mass, rs≈2·6.674·10−11·1.989·1030/(2.998·108)2≈2.95 km.
Solution path
Substitute quantities in the documented order, keep units consistent and compare the result with the check stated in the example.
The equation describes a specific relation among astronomical quantities; it supports a reliable conclusion only when measurements match the variables’ definitions. Start by identifying each quantity, unit and reference frame. The relation rs=\frac{2GM}{c^2} organizes those values mathematically, while the physical model determines whether it applies to the situation. A number alone does not establish that. Record assumptions, signs and any approximations. This keeps separate what follows directly from calculation and what requires interpretation. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
The relevant limits are stated in the subject background: The Einstein field-equation solution for a spherical, nonrotating, uncharged mass has horizon rs=2GM/c2. A rotating object requires Kerr geometry, so this is not a complete horizon description. The radius is not a material surface. Knowing a mass and radius alone does not make an object a black hole; matter must be concentrated inside the horizon. Read them before substitution because ideal geometry, small angles, constant properties or a particular cause of motion may be part of the derivation. Measurement uncertainty is also distinct from model validity: a precise measurement does not make an unsuitable approximation correct. Check that quantities use consistent conventions. If the assumptions fail, use a more suitable model or explicitly report the result as an estimate rather than an exact measurement. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
Use the documented worked example as a calculation check: For M equal to one solar mass, rs≈2·6.674·10−11·1.989·1030/(2.998·108)2≈2.95 km. Carry units through the arithmetic rather than handling numbers alone. Do they combine into the unit of the requested quantity? A dimensional mismatch often reveals a conversion error or swapped variables. If possible, substitute the result back into a rearranged original relation. Finally compare its order of magnitude with familiar astronomical scales; plausibility is not proof, but it is a useful error check. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
Read the exponents and placement of quantities in rs=\frac{2GM}{c^2}: direct proportionality behaves differently from an inverse or power relation. Mentally double one input at a time and infer how the requested quantity changes. In observational data, a small relative uncertainty can matter greatly when a relation uses a small angle, a difference or an inverse quantity. Distinguish this sensitivity from measurement precision: the equation describes error propagation, not how well an instrument measures. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
State the numerical value with its unit, the input measurements and the main model assumption. If using an approximation, label the result as an estimate and say which effects were ignored. Avoid conclusions beyond the equation: a geometric or photometric quantity is not automatically a complete description of a star or galaxy. When measurement errors are known, do not report more precision than the data support. This concise record makes both calculation and interpretation traceable and reproducible. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
Quanta exam set
Curated exam set for "Schwarzschild radius of a nonrotating black hole":
Question (front)
Which formula describes Schwarzschild radius of a nonrotating black hole?
Answer in your set
Question (front)
How do you rearrange Schwarzschild radius of a nonrotating black hole for Defining relation?
Answer in your set
Question (front)
Which common mistake happens with Schwarzschild radius of a nonrotating black hole?
Answer in your set
+ 7 more cards: units, variables, derivation, example, exam task
The formula set contains 10 cards. After importing, review them and start practicing; your ratings determine later review intervals.
This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.
Create a curated FSRS exam set for rs = 2GM/c²: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.
Curated formula set · LaTeX · FSRS spaced repetition · import from Essential