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Special Relativity

General Relativity

Schwarzschild's Solution and Black Hole

Kerr's Solution and Rotating Black Hole

A Scenario for Time Travel

Hawking Radiation

Black Hole and Information

Standard Cosmology

Cosmological Constant and de Sitter Universe

Theory of Cosmic Inflation and Acceleration

Static Universe

Angular-Size Redshift Relation

Euclidean Space

Five Dimensional Space-time

Gravitational Wave

Time

Un-relativistic Theory

Momentum Space

Time Reborn

Classical mechanics describes the way objects move and interact in accordance with Newton's laws of motion. The basic assumptions involve a frame of reference (x,y,z) with respect to which object with mass m moves, there is an independent time variable t to record the sequence of the movement, the gravitational or electromagnetic interaction between objects is instantaneous, and objects with geometric extent are often idealized as a point (with the justification that the size is much smaller than the distance involved). The basic equation is: | |

## Figure 01a Newton's 3 Laws |

This formula is known as the equation of motion and looks deceptively simple. However, the force

F

which are essentially three separate differential equations. Strictly speaking, Eq.(2) is applicable only to a point mass without spatial extent. But it is often used on extended objects such as a brick (Diagram c, Figure 01a), the Earth, ... without stating explicitly the idealization. It has created lot of confusion in countless inquiring minds, many of which have eventually developed a phobia for physics. The simplification is valid only if the distance scale is much larger than the size of the object(s). The same kind of problem also occurs in the Big Bang theory which proposes the origin of the universe from a point with infinite density, and in the theory of elementary particles, which is plagued with infinities - the result of treating the particles as points without internal structure.

The force

The Newton's third law is (in his own words): "

The Newton's law of universal gravitation is in the form:

where G is the gravitational constant, m_{1} and m_{2} are the masses of the two objects interacting via gravitation, r is the distance between these two objects, and (r/r) is an unit vector along the direction of r (see Figure 01b).If one of the objects is much heavier than the other, e.g., m _{1} >> m_{2} like the Sun / Earth system, then m_{1} can be placed in the origin of the coordinate system and Eq.(1) can be solved as a one-body problem. In case the two masses are similar, the problem can be reduced to a one-body problem with a fictitious object moving around the center of mass, and Eq.(1) is still applicable. The equation of motion becomes rapidly un-manageable for system of three bodies and beyond. Eq.(1) would be applied to all the objects and the force on one object would involve the interaction with all the others. This is the situation often encountered in celestial mechanics with spacecraft flying among planets. The solution is usually obtained by some kind of approximation and by numerical computation using large | |

## Figure 01b Gravitational Interaction |
computers. See Newton's Laws in cartoons. |

x' = x - Vt, y' = y, z' = z, and t' = t ---------- (4)

It is obvious that the length

dl^{2} = dx^{2} + dy^{2} + dz^{2} = dx'^{2} + dy'^{2} + dz'^{2} ---------- (5)According to Eq.(4) if the velocity of light (in the x direction) is c in the S frame it would be c' = c - V in the S' (moving) frame. In classical mechanics, the "Absolute Frame of Reference" is a hypothetical entity identified as the frame of reference with the origin at the center of mass of system of fixed stars. Only with respect to this absolute frame of reference would the velocity of light equal to c = 3x10 ^{10} cm/sec. | |

## Figure 02a Galilean Transformation [view large image] |
It was later suggested that the medium in which light propagates - the ether - would be an even better absoute frame of reference. |

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