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Some fields associated with a moving point, and weak solutions of the wave equation

Published online by Cambridge University Press:  24 October 2008

G. N. Ward
G. N. Ward
Affiliation:
The College of Aeronautics, Cranfield
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Extract

If P is a point in space-time, and P′ is a point whose space coordinates are continuous functions of time and whose speed in space is always less than the speed of light, then the time, T, at which the square of the interval between P and P′ is R, is a two-valued real function of R and the space-time coordinates of P for a range of R that includes R = 0. For fixed R in this range, each branch of T defines a scalar field over the whole of space-time, and, as R → 0, these fields reduce to the retarded and advanced times of P′ at P. Any function of T is also a field that reduces to a retarded or advanced field as R → 0. An interesting property of these fields is that there are identical relations between their derivatives with respect to R and the space-time variables, which enable the space-time derivatives to be expressed in terms of the R-derivatives; moreover, these relations are linear in the derivatives and take the same form for every such field.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 1 , January 1965 , pp. 191 - 199
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

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