Lecture 28 Lecture Notes

Lecture 28 Lecture Notes - properties. The form invariance...

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8.4 General transformation rules Matrix translation rules: use free reordering of index notation - use transposition, e.g. - first index is row index second index column vector (upper or lower position is just reminder of transformation rules) remember: => bring identical indices next to each other by this rule Examples: (from above) 8.5 Transformation rules for derivatives Define: Claim: transforms as covariant vector! Transformation and transformation inverse to Lec 28 - 18. Mar March 17 09 9:28 PM Lecture Notes Page 1
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8.6 Raising and lowering operation (as before) => can be used to lower indices Raising of indices? Define with matrix representation Then is the inverse of Consistency: (symmetry) works also for derivatives: 8.7 Electrodynamics in relativistic formulation In order to do a relativistic formulation of electrodynamics, we demand form invariance of the laws of electrodynamics in all inertial reference frames. The basic trick is then simple: find a way to express the laws with 4- vectors such that all elements except one have known transformation
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Unformatted text preview: properties. The form invariance then implies the transformation property of the remaining 4-vector object. This leads to expression for the source terms, combining and , And also for the potentials and These form the basic elements of the relativist theory, and we then derive new objects containing the e/m fields, the energy etc. 8.7.1 Continuity equation => should hold in all inertial coordinate systems use co-variant derivative vector transforms as scalar must transform as contra-variant vector! Lecture Notes Page 2 4-vector current and Example: current in resting conductor: moving balanced net charge in moving frame: => moving observer sees non-zero net charge density! confirms again: charge density is not a scalar 8.7.2 Potentials (in Lorentz gauge) gauge condition note that with transforms as scalar! must transform as contra-variant vector! 4-vector potential Gauge condition: Lecture Notes Page 3...
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Lecture 28 Lecture Notes - properties. The form invariance...

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