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Relativity, Cosmology, and Time

Special Relativity

Michelson-Morley Experiment The search was on in the late 19th Century for the elusive ether. Finally in 1887 Michelson and Morley demonstrated conclusively that the speed of light in different inertial frames is the same everywhere (Figure 02b). The Maxwell's equations for electromagnetic wave also indicates that the speed of light is a constant regardless of the relative motion of the person measuring that speed. But as just mentioned above, the velocity of light is different in different inertial frame according to the Galilean transformation. The theory of special relativity was postulated to reconcile this kind of inconsistency. The theory does away with the idea of absolute frame of reference such as the ether, and treats all inertial frames on an equal footing.

Figure 02b Michelson-Morley Experiment [view large image]

Constant Light Speed Mathematically if a spherical light wave is generated at the origin of the S and S' inertial frames when they are coincided at t = 0, the statement about the constant velocity of light in different inertial frames can be expressed as:

x2 + y2 + z2 = c2 t2   or   x2 + y2 + z2 - c2 t2 = 0    (for an observer in the S frame) ---------- (6)
x'2 + y'2 + z'2 = c2 t'2   or   x'2 + y'2 + z'2 - c2 t'2 = 0     (for an observer in the S' frame) ---------- (7)

Figure 02c Constant Speed of Light
[view large image]

This situation is possible only when t is not equal to t'. It can be shown that the Lorentz transformation below would satisfy the requirement for Eq.(6) and (7) for a S' frame observer viewing the S frame :

x' = (x - Vt) / (1 - V2/c2)1/2,   y' = y,   z' = z,   and   t' = (t - Vx/c2) / (1 - V2/c2)1/2 ---------- (8a)

The inverse transformation for a S frame observer viewing the S' frame is :

x = (x' + Vt') / (1 - V2/c2)1/2,   y = y',   z = z',   and   t = (t' + Vx'/c2) / (1 - V2/c2)1/2 ---------- (8b)

where V denotes the velocity between inertial frames, and v (in the subsequent text) would be the velocity of a point in an inertial frame.

Eq.(8a) reduces to the Galilean transformation in Eq.(4) when V/c << 1. Figure 02c shows pictorially the wave fronts in the S and S' frames from the perspective of an observer in the rest frame. In this way, x, y, z, and ct form a four-dimensional space known as the Minkowski space-time. It revolutionizes high-energy physics when velocity of the particles is close to the velocity of light (relative to the observer's inertial frame). All the basic formulae such as the field equations and the Lagrangians have to be invariant under the Lorentz transformation, and the definition for many physical entities changes form at high speed as listed in the followings:

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