The light cone calculus is remarkably similar in theorem to Slide 11. (01)
"Null geodesics lie along a dual-cone defined by the equation (02)
ds^2 = 0 = dx_1^2 + dx_2^2 - c^2 dt^2 (03)
or (04)
dx_1^2 + dx_2^2 = c^2 dt^2 (05)
Which is the equation of a circle with r=c*dt. If we extend this to three
spatial dimensions, the null geodesics are continuous concentric spheres, with
radius = distance = c×(±time). (06)
ds^2 = 0 = dx_1^2 + dx_2^2 + dx_3^2 - c^2 dt^2 (07)
dx_1^2 + dx_2^2 + dx_3^2 = c^2 dt^2 (08)
This null dual-cone represents the "line of sight" of a point in space. That
is, when we look at the stars and say "The light from that star which I am
receiving is X years old.", we are looking down this line of sight: a null
geodesic. We are looking at an event d = \sqrt{x_1^2+x_2^2+x_3^2} meters away
and d/c seconds in the past. For this reason the null dual cone is also known
as the 'light cone'. (The point in the lower left of the picture below
represents the star, the origin represents the observer, and the line
represents the null geodesic "line of sight".) (09)
The cone in the −t region is the information that the point is 'receiving',
while the cone in the +t section is the information that the point is 'sending'. (010)
The geometry of Minkowski space can be depicted using Minkowski diagrams, which
are also useful in understanding many of the thought-experiments in special
relativity." (011)
Great stuff!!! (012)
Duane (013)
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Adobe Systems, Inc. - http://www.adobe.com
Vice Chair - UN/CEFACT http://www.uncefact.org/
Chair - OASIS SOA Reference Model Technical Committee
Personal Blog - http://technoracle.blogspot.com/
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