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Chapter 2
GENERAL THEORY OF
RELATIVITY
- PHYSICS OF GRAVITY
2.1. Acceleration and gravity from the point
of view of special theory of relativity
2.2. Versatility
- a basic property and the key to understanding the nature of
gravity
2.3. The
local principle of equivalence and its consequences
2.4. Physical
laws in curved spacetime
2.5. Einstein's
equations of the gravitational field
2.6. Deviation
and focus of geodesics
2.7. Gravitational
waves 2.8. Specific
properties of gravitational energy
2.9.Geometrodynamic system of units
2.10. Experimental
verification of the theory of relativity and gravity
2.9. Geometrodynamic system of units
The basic benefit of the theory of relativity is the knowledge that the positional coordinates and temporal moments of events do not have a separate invariant meaning, that space and time form a unified space-time continuum. Only "distances" measured in space-time, ie space-time intervals, are of absolute importance. For this points of wiev is seems inconsistent, that the distance (interval) in space-time are measures in different directions in different measuring units: in the spatial directions in meters, along the time axis in seconds. It is similar to measuring distances here on earth in the horizontal direction (ie length and width) in meters, while measuring distances in the vertical direction (height and depth) in inches. This would create unnecessary complications when measuring lengths in "oblique" directions - determining a distance of points with different horizontal coordinates lying variously high.
Just as distances in space are naturaly measured in all directions using the same units, it is reasonable to measure "distances" in space-time, i.e. space-time intervals, in all directions by the same units. If we take one meter (a unit of length in space) as a basis, it is advantageous to mark the time axis t so that one scale division represents the distance that travels light in one second, so use the coordinate x° = ct º t*, as we have already they did in space-time diagrams. In other words, we measure length and time in meters, placing the speed of light, which is the best standard of speed, equal to one: c = 1 (one meter of length in "one meter" of time). The geometrometric time t* is related to the ordinary time t by the relation
t * _{[m]} = c . t _{[s]} @ 3. 10 ^{8} . t _{[s]} . | (2,107) |
For space-time measurements, we only need one unit - a meter.
Another basic physical quantity is mass. In Newtonian physics, matter has two basic manifestations - inertia and gravity, where these manifestations are independent, the equality (proportionality) of inertial and gravitational mass is an empirical fact that cannot be explained in theory. To quantitatively express the amount of matter another quantity was determined - mass, weight - independent of the quanties of length and time, the unit has been chosen kilogram.
In the general theory of relativity, which blurs the distinction between inertia and gravity, the mass of bodies is measured by their gravitational manifestations, that is, by how they curve space-time. In other words, mass can be determined by purely geometric measurements in spacetime - by measuring length. The gravitational action of a body of mass M according to Newton's theory (which is the limit of GTR) gives each test particle located at a distance r an acceleration a º d^{2}r/dt^{2} = G.M/r^{2}, which in geometrodynamic units can be rewritten a* s d^{2}r*/dt*^{2} = (G·M/c^{2})/r*^{2}. For the geometrodynamic mass, it is then natural to choose a quantity with a length dimension
M * _{[m]} = (G / c ^{2} ). M _{[kg]} @ 0.743. 10 ^{-27} . M _{[kg]} . | (2.108) |
In geometrodynamic units, for example, the mass of the Sun M_{¤} @ 1.48 km, the mass of the Earth M_{l} @ 0.44 cm, the average person "weighs" about 5.10^{-26} m. That kilograms and meters are two equivalent units for measuring the mass of bodies (ie the measures of their inertia and gravity) can also be seen from the fact that the gravitational radius of a body r_{g} = 2G.M/c^{2} = 2.M* (see §3.4) is a completely unambiguous characteristic of the structure of spacetime around a gravitational body equivalent to its mass M (expressed in kilograms).
Quantities describing electromagnetic field can also be designed purely geometric method based on the curvature R_{ik}, by this field (included therein energy-momentum) induces in spacetime. From the dimensional analysis of the coupled Einsten-Maxwell equations, it follows that the geometrodynamic units of electric and magnetic field intensity are related to ordinary units by the relation
F *_{ik }_{[cm} -1 _{]} = (G ^{1/2} / c ^{2} ) F _{ik }_{[cm}1/2 _{.g}1/2 _{.s} -1 _{]} @ (2,874.10 ^{-25} cm ^{-1} / Gauss) F _{ik} .
From Maxwell's equations (2.32), as well as from the generalized Lorentz equation (2.30) of the motion of a charge in an electromagnetic field, it follows that the same conversion relationship connects the geometrodynamic measure of an electric charge with its ordinary units:
Q *_{[cm]} = (G ^{1/2} / c ^{2} ). Q @ (2,874.10 ^{-25} cm / CGSE). Q _{[cm}3/2 _{.g}1/2 _{.s} -1 _{]} ,
or for SI units
Q * _{[m]} @ 0.862 . 10 ^{-17} . Q _{[Coulomb]} . | (2,109) |
Geometrodynamics system of units G = c = 1 indicates "tightened to the end" ideas about the relation of space, time, gravitation and mass in general relativity. Matter should be measured using its most characteristic manifestations (symptoms). What most characterizes all forms of matter is the universal gravitational action, the curvature of spacetime. From this point of view, therefore, all physical measurements are, after all, geometric measurements in space-time, for which we only need one unit - a unit of length. In the words of Misner and Wheeler [180]: physics is geometry.
Einstein's equations written in geometrodynamic units do not contain any constant :
G_{ik} s R_{ik} - ^{1}/_{2} g_{ik} R = 8 p T_{ik} . | (2,110) |
The equation of motion of a test particle of mass m with charge q in an electromagnetic and gravitational field is
m* [d^{2}x^{i}/dt^{2} + G^{i}_{kl}(dx^{k}/dt)(dx^{l}/dt)] = q* F*^{i}_{k}dx^{k}/dt . | (2,111) |
The first pair of Maxwell's equations have the same shape in geometric units as in ordinary units
F*_{ik,l} + F*_{li,k} + F*_{kl,i} = 0 , ie. div H* = 0 , rot E* + ¶E*/¶t* = 0 , | (2.112a) |
the second pair of Maxwell's equations in the geometrodynamic system is
F*^{ik}_{;k} = 4p j*^{ i} , ie. div E* = 4p r* , rot H* - ¶H*/¶t* = 4p j* . | (2.112b) |
Finally, the coupled Einstein-Maxwell equations describing the behavior of the system [free electromagnetic field + gravitational field excited by it] have the form in geometrodynamic units :
R^{i}_{k} - ^{1}/_{2} d^{i}_{k} R = 2 *F^{l}_{k}*F^{i}_{l} - ^{1}/_{2} d^{i}_{k} *F^{l}_{m}*F^{m}_{l} . | (2,113) |
The conversion coefficients between ordinary (SI) and geometrodynamic (*) units for the basic quantities of classical physics are given in the following table :
The use of a geometrodynamic system of units is very advantageous in the general theory of relativity, because it greatly simplifies the notation of most relations, in which the constants G and c no longer overshadow the content connections between different quantities and their relations to the structure of spacetime. In the following, therefore, we will work in these geometrodynamic units, while we will no longer mark the relevant symbols with asterisks. However, we will also state important final relations in the form corresponding to conventional units.
Gravity, black holes and space-time physics : | ||
Gravity in physics | General theory of relativity | Geometry and topology |
Black holes | Relativistic cosmology | Unitary field theory |
Anthropic principle or cosmic God | ||
Nuclear physics and physics of ionizing radiation | ||
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