For this reason the null dual cone is also known as the'light cone'. ( The point in the lower left of the picture above right represents the star, the origin represents the observer, and the line represents the null geodesic " line of sight " .)
42.
In these coordinate systems, outward ( inward ) traveling radial light rays ( which each follow a null geodesic ) define the surfaces of constant " time ", while the radial coordinate is the usual area coordinate so that the surfaces of rotation symmetry have an area of " r " 2 } }.
43.
If we study the past light cone of a given observer, we find that null geodesics moving orthogonally to \ partial _ y " spiral inwards " toward the observer, so that if he looks " radially ", he sees the other dust grains in progressively " time-lagged positions ".
44.
Plotting the " tracks " of some representative null geodesics through a given event ( that is, projecting to the hyperslice \ scriptstyle t \, = \, 0 ), we obtain a picture which looks suspiciously like the family of all semicircles through a point and orthogonal to the Rindler horizon . ( See the figure .)
45.
As we can see from the figure illustrating the Rindler wedge, the locus " x " = 0 in the Rindler chart corresponds to the locus " T " 2 = " X " 2, " X " > 0 in the Cartesian chart, which consists of two null half-planes, each ruled by a null geodesic congruence.
46.
Under Will's presentation ( which was inspired by John Lighton Synge's interpretation of the theory ), Whitehead's theory has the curious feature that electromagnetic waves propagate along null geodesics of the physical spacetime ( as defined by the metric determined from geometrical measurements and timing experiments ), while gravitational waves propagate along null geodesics of a " flat background " represented by the metric tensor of Minkowski spacetime.
47.
Under Will's presentation ( which was inspired by John Lighton Synge's interpretation of the theory ), Whitehead's theory has the curious feature that electromagnetic waves propagate along null geodesics of the physical spacetime ( as defined by the metric determined from geometrical measurements and timing experiments ), while gravitational waves propagate along null geodesics of a " flat background " represented by the metric tensor of Minkowski spacetime.
48.
I'm too lazy to draw a picture illustrating this " strong lensing " effect, so I probably shouldn't have said anything at all . . . oh well . . . For physics students, you can figure out what I'm talking about from what you read about null geodesics in Chrandrasekhar, " Mathematical Theory of Black Holes " .---CH 03 : 03, 24 July 2006 ( UTC)
49.
I can prove that to build a finite time machine, you need negative energy . " This result comes from Hawking's 1992 paper on the chronology protection conjecture, where he examines " the case that the causality violations appear in a finite region of spacetime without curvature singularities " and proves that " [ t ] here will be a Cauchy horizon that is compactly generated and that in general contains one or more closed null geodesics which will be incomplete.
50.
For a metric of the general form of the drainhole metric, with \ partial _ t + c f ( \ rho ) \ partial _ \ rho as the velocity field of a flowing ether, the coordinate velocities d \ rho / dt of radial null geodesics are found to be c ( f ( \ rho ) + 1 ) for light waves traveling against the ether flow, and c ( f ( \ rho )-1 ) for light waves traveling with the flow.
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