Geometrized unit system
A geometrized unit system [1]: 36 or geometrodynamic unit system is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum, c, and the gravitational constant, G, are set equal to unity.
The geometrized unit system is not a completely defined system. Some systems are geometrized unit systems in the sense that they set these, in addition to other
This system is useful in
Many equations in relativistic physics appear simpler when expressed in geometric units, because all occurrences of G and of c drop out. For example, the
Practical measurements and computations are usually done in
Definition
Geometrized units were defined in the book
In geometric units, every time interval is interpreted as the distance travelled by light during that given time interval. That is, one second is interpreted as one light-second, so time has the geometric units of length. This is dimensionally consistent with the notion that, according to the kinematical laws of special relativity, time and distance are on an equal footing.
Energy and momentum are interpreted as components of the four-momentum vector, and mass is the magnitude of this vector, so in geometric units these must all have the dimension of length. We can convert a mass expressed in kilograms to the equivalent mass expressed in metres by multiplying by the conversion factor G/c2. For example, the Sun's mass of 2.0×1030 kg in SI units is equivalent to 1.5 km. This is half the Schwarzschild radius of a one solar mass black hole. All other conversion factors can be worked out by combining these two.
The small numerical size of the few conversion factors reflects the fact that relativistic effects are only noticeable when large masses or high speeds are considered.
Conversions
Listed below are all conversion factors that are useful to convert between all combinations of the SI base units, and if not possible, between them and their unique elements, because ampere is a dimensionless ratio of two lengths such as [C/s], and candela (1/683 [W/sr]) is a dimensionless ratio of two dimensionless ratios such as ratio of two volumes [kg⋅m2/s3] = [W] and ratio of two areas [m2/m2] = [sr], while mole is only a dimensionless
| m | kg | s | C | K | |
|---|---|---|---|---|---|
| m | 1 | c2/G [kg/m] | 1/c [s/m] | c2/(G/(ε0))1/2 [C/m] | c4/(GkB) [K/m] |
| kg | G/c2 [m/kg] | 1 | G/c3 [s/kg] | (Gε0)1/2 [C/kg] | c2/kB [K/kg] |
| s | c [m/s] | c3/G [kg/s] | 1 | c3/(G/(ε0))1/2 [C/s] | c5/(GkB) [K/s] |
| C | (G/(ε0))1/2/c2 [m/C] | 1/(Gε0)1/2 [kg/C] | (G/(ε0))1/2/c3 [s/C] | 1 | c2/(kB(Gε0)1/2) [K/C] |
| K | GkB/c4 [m/K] | kB/c2 [kg/K] | GkB/c5 [s/K] | kB(Gε0)1/2/c2 [C/K] | 1 |
Geometric quantities
The components of curvature tensors such as the
Path curvature is the reciprocal of the magnitude of the
Any
Physical quantities such as
Here is a table collecting some important physical quantities according to their dimensions in geometrized units. They are listed together with the appropriate conversion factor for SI units.
| Quantity | SI dimension | Geometric dimension | Multiplication factor |
|---|---|---|---|
| Length | L | L | 1 |
| Time | T | L | c |
| Mass | M | L | G c−2 |
| Velocity | L T−1 | 1 | c−1 |
| Angular velocity | T−1 | L−1 | c−1 |
| Acceleration | L T−2 | L−1 | c−2 |
| Energy | M L2 T−2 | L | G c−4 |
| Energy density | M L−1 T−2 | L−2 | G c−4 |
| Angular momentum | M L2 T−1 | L2 | G c−3 |
Force
|
M L T−2 | 1 | G c−4 |
| Power | M L2 T−3 | 1 | G c−5 |
| Pressure | M L−1 T−2 | L−2 | G c−4 |
| Density | M L−3 | L−2 | G c−2 |
| Electric charge | T I | L | G1/2 c−2 ε0−1/2 |
| Electric potential | M L2 T−3 I−1 | 1 | G1/2 c−2 ε01/2 |
| Electric field | M L T−3 I−1 | L−1 | G1/2 c−2 ε01/2 |
| Magnetic field | M T−2 I−1 | L−1 | G1/2 c−1 ε01/2 |
This table can be augmented to include temperature, as indicated above, as well as further derived physical quantities such as various moments.
References
- ^ ISBN 978-0-7167-0344-0.
- S2CID 235581301.
- ISBN 0-226-87033-2. See Appendix F