Einsteins Theory of Relativity 3 Electrodynamics Ulrich R

Einstein‘s Theory of Relativity 3. Electrodynamics Ulrich R. M. E. Geppert 9/26/2020 U. R. M. E. Zielona Gora 1

since c is independent of IS⇒ Maxwell equations are valid in all IS Ludwig Boltzmann, introduction to Maxwell‘s equations: „was it a god who wrote these symbols? “ (a Goethe quote) 9/26/2020 U. R. M. E. Zielona Gora 2

Maxwell equations obey already Einstein‘ s principle of relativity reminder: All laws of physics hold in all ISs ⇒ GT replaced by LT Maxwell‘s equations have to be re-written by use of 4 -vectors and 4 -tensors in such a way that their relativistic invariance and the behaviour of fields, currents, and charge distributions with respect to Lorentz transformations become transparent 9/26/2020 U. R. M. E. Zielona Gora 3

Reminder: vector and scalar potential, Lorenz gauging not Hendrik Lorentz but Ludvig Lorenz „Ampere“ inhomogeneous MG: „Gauss“ homogeneous MG: „Faraday“ 9/26/2020 U. R. M. E. Zielona Gora 4

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rewrite Ampere‘s and Gauss‘ law in terms of potentials „Lorenz condition“ 9/26/2020 U. R. M. E. Zielona Gora 6

apply gauge transformation to potentials in Lorenz condition: 9/26/2020 U. R. M. E. Zielona Gora 7

Ampere‘s and Gauss‘ law in terms of potentials: „wave equations“ form a set of equations completely equivalent to Maxwell‘s equations. 9/26/2020 U. R. M. E. Zielona Gora 8

4 -potential 9/26/2020 U. R. M. E. Zielona Gora 9

4 -current 9/26/2020 U. R. M. E. Zielona Gora 10

wave eqs. , Lorenz condition (~MG) in manifestly covariant forms: 9/26/2020 U. R. M. E. Zielona Gora 11

e. g. explicitely for the x-components 9/26/2020 U. R. M. E. Zielona Gora 12

introduce the electromagnetic field tensor its covariant form 9/26/2020 U. R. M. E. Zielona Gora 13

dual field strength tensor totally anti-symmetric Levi-Civita (pseudo-)tensor relations between forms of the field strength tensor (exercises? ) 9/26/2020 U. R. M. E. Zielona Gora 14

covariant form of inhomogeneous Maxwell equations: 9/26/2020 U. R. M. E. Zielona Gora 15

covariant form of homogeneous Maxwell equations: 9/26/2020 U. R. M. E. Zielona Gora 16

covariant representation of the Lorentz force: 4 -momentum charge 4 -velocity proper time 9/26/2020 U. R. M. E. Zielona Gora 17

energy momentum tensor of the electromagnetic field: 9/26/2020 U. R. M. E. Zielona Gora 18

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by use of the field strength tensor: 9/26/2020 U. R. M. E. Zielona Gora 20

in case of non-conservation: is the force density exerted by the elm field onto the current distribution 9/26/2020 U. R. M. E. Zielona Gora 21

we receive informations about stars via electromagnetic waves travelling through space ⇒ electromagnetic waves in vacuum (reminder): solution of wave equations: plane waves complex amplitudes insert solution into wave equation: ⇒ dispersion relation: 9/26/2020 U. R. M. E. Zielona Gora 22

insert solutions into Mawell‘s equations: 9/26/2020 U. R. M. E. Zielona Gora 23

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Frequency and wavelength of an emitter in motion decreases or increases depending on the direction of the motion. „redshift“ or „blueshift“ 9/26/2020 U. R. M. E. Zielona Gora 25

Doppler effect 1. longitudinal 9/26/2020 U. R. M. E. Zielona Gora 26

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not as large as the longitudinal one „aberration“ 9/26/2020 U. R. M. E. Zielona Gora 28

astrophysical evidences cosmological redshift - light from distant galaxies, supernovae is observed redshifted on Earth - the more distant the more redshifted ⇒ accelerated espansion of the universe (Hubble) - for not too large distances returns the longitudinal Doppler effect correct results however - long. DE is calculated in special relativity, i. e. assumes IS in which both emitter and observer have defined positions and velocities later 9/26/2020 U. R. M. E. Zielona Gora 29

⇒ velocity of galaxies, stars, SN… expanding universe 9/26/2020 U. R. M. E. Zielona Gora 30

Doppler effect seen in radiation from millisecond pulsars (MSP) only a theoretical example to demonstrate DE: 9/26/2020 U. R. M. E. Zielona Gora 31

redshift: 0. 05, blueshift: -0. 05 9/26/2020 U. R. M. E. Zielona Gora 32

take this only as an illustrative example and not as a correct description of the red-and blueshifted spectral lines because this SR Doppler effect is superimposed by the gravitational redshift (GR) that is the dominating one if the emitting region is close to the pole of the neutron star… later 9/26/2020 U. R. M. E. Zielona Gora 33