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A fuller explanation of the concept of coordinate time arises from its relations with proper time and with clock synchronization. Synchronization, along with the related concept of simultaneity, has to receive careful definition in the framework of general relativity theory, because many of the assumptions inherent in classical mechanics and classical accounts of space and time had to be removed.
which connects all the 3 spatial coordinates of that particle together, so they are not independent. The constraint may change with time, so time t will appear explicitly in the constraint equations. At any instant of time, any one coordinate will be determined from the other coordinates, e.g. if x k and z k are given, then so is y k.
In relativity, proper time (from Latin, meaning own time) along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time, which is independent of coordinates, and is a Lorentz scalar . [ 1 ]
In physics, sometimes units of measurement in which c = 1 are used to simplify equations. Time in a "moving" reference frame is shown to run more slowly than in a "stationary" one by the following relation (which can be derived by the Lorentz transformation by putting ∆x′ = 0, ∆τ = ∆t′):
This equation will often depend on temperature, so a heat transfer equation is required or the postulate that heat transfer can be neglected. Next, notice that only 10 of the original 14 equations are independent, because the continuity equation T a b ; b = 0 {\displaystyle T^{ab}{}_{;b}=0} is a consequence of Einstein's equations.
The equation of time is obtained from a published table, or a graph. For dates in the past such tables are produced from historical measurements, or by calculation; for future dates, of course, tables can only be calculated. In devices such as computer-controlled heliostats the computer is often programmed to calculate the equation of time.
For typical ionic solids, the cations are smaller than the anions, and each cation is surrounded by coordinated anions which form a polyhedron.The sum of the ionic radii determines the cation-anion distance, while the cation-anion radius ratio + / (or /) determines the coordination number (C.N.) of the cation, as well as the shape of the coordinated polyhedron of anions.
The spherical coordinate system is commonly used in physics. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distance r, polar angle θ , and azimuthal angle φ . The symbol ρ is often used instead of r.