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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] |
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 |
L = x2 - x1 = L' / (1 - V2/c2)1/2 or L' = L (1 - V2/c2)1/2, which is known as the Lorentz contraction. The length of a moving rod is shorter according to an observer in the S' frame. The phenomena is demonstrated in high energy collision, when the round shape of atom becomes a pancake with the flatten face perpendicular to the direction of motion (Figure 03). | |
Figure 03 Lorentz Contraction from S' Frame [view large image] |
c2 dt2 - dx2 - dy2 -dz2 = 0 ---------- (9) This is called the null line for object moving at the speed of light. In place of the invariant lenght d in Eq.(5) under the Galilean Transformation, for object moving with velocity not equal to the speed of light, the space-time interval "ds" (aka the "world line" - the path of a particle in 4-D space-time) is : ds2 = c2 dt2 - dx2 - dy2 - dz2 ---------- (10) which is invariant under the Lorentz transformation. For a clock at rest in a reference frame, that is, with dx = dy = dz = 0, d = ds/c is called the proper time. | |
Figure 04c Minkowski Space-time [view large image] |
In general, by denoting v2 = [(dx/dt)2 + (dy/dt)2 + (dz/dt)2], Eq.(10) can be re-written as : ds2 = (1-v2/c2)c2dt2. |
The interval is called time-like if ds2 0. As shown in figure 04c, event 1 can be related causally to an observer at the origin x = 0, ct = 0 (the "now") from the past with incoming velocity v c. If ds2 < 0, it is called space-like, which implies event 2 is entirely unrelated to that observer, since it would require a flow of information at speed faster than the speed of light. Similarly, time-like separation can influence a receiver in the future, while outgoing communication is impossible with space-like linkage. The space-time interval ds (or proper time d) plays the role of the time parameter in Newtonian mechanics to keep track of the development of events as shown in the generalized equation of motion in Eq.(12b). Note that the velocity of light c is constant in all Lorentz frame of references in Figure 05 as originally envisioned. | |
Figure 05 Minkowski Space-time Transformation [view large image] |
BTW, an event is a point in the four dimension space-time. |
t'2 - t'1 = [(x1 - x2)(V/c2)] / (1 - V2/c2)1/2 Figure 05a shows the different perspective according to the stationary observer C, who sees simultaneity of two lightnings, while D in the moving bus detects the lightnings in different time. | |
Figure 05a Simultaneity [view large image] |
Specifically in the moving bus, the signal from A takes longer time t'1 to reach D than t'2 from B. |
(v/c)2 = 1/[1 + (moc/p)2] or p2 = mo2c2{(v/c)2/[1 - (v/c)2]} For mo2 > 0, the particle is always moving slower than the velocity of light. For mo2 = 0, v is always equal to c. If mo2 < 0 then v > c, i.e., the particle is moving faster than the velocity of light. Such particle is called tachyon, which has the peculiar property that it slows down to approach the velocity of light with increasing momentum p, while its speed goes up to infinity as p falls to |mo|c (see Figure 06a). | |
Figure 06a Mass and Velocity [view large image] |
There is no direct evidence that tachyons exist, and most physicists believe there is something wrong or it requires some sort of interpretation when they appear in a theory. The spontaneous symmetry breaking potential in the Higgs mechanism is one example of tachyon in physics. |
Mathematical Entity | Timelike | Spacelike | Spacelike Property |
---|---|---|---|
4-D Spacetime Vector | x = (ct, x) | x = (ct, x) | Outside light cone |
Spacetime Interval | ds2 = c2dt2 - dx2 0 | ds2 = c2dt2 - dx2 < 0 | v > c - tachyon |
4-Momentum | p = (E/c, p) | p = (E/c, p) | Outside light cone |
4-Momentum Squared | mo2c2 = p2 = E2/c2 - p2 0 | mo2c2 = p2 = E2/c2 - p2 < 0 | Off mass-shell - tachyon |
4-Momentum Transfer | p' - p = [(E' - E)/c, (p' - p)] = q | p' - p = [(E' - E)/c, (p' - p)] = q | Outside light cone |
4-Momentum Transfer Squared | (p' - p)2 = (E' - E)2/c2 - (p' - p)2 = q2 0 | (p' - p)2 = (E' - E)2/c2 - (p' - p)2 = q2 < 0 | Off mass-shell |
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Figure 06b Special Theory of Relativity - |
Age-of-Travelling-Twin = T'(outbound) + T'(inbound) = T(1 - |V|/c) + T(1 + |V|/c) = 2T = Age-of-Earth-bound-Twin (see "Twin Paradox"). |