Una teoria consistente dell'induzione...

Una teoria consistentedell’induzione

elettromagnetica

Giovanni Romano

Electromagnetics

Early discoveries andtheory

Simplifications

Early relativity theory

New theory

Una teoria consistente dell’induzioneelettromagnetica

Giovanni Romano

Accademia di Scienze Fisiche e Matematiche in Napoli

1 febbraio 2013


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

ED=Electrodynamicsand

DG=Differential Geometry

ED is an important source of inspiration for DG and DG is thenatural tool to develop a mathematical modeling of ED

Founders of DG


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory




Founders of DG

Johann Carl Friederich Gauss (1777 - 1855)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory




Founders of DG

Hermann Gunther Grassmann (1809 - 1877)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory




Founders of DG

Bernhard Riemann (1826 - 1866)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory




Founders of DG

Marius Sophus Lie (1842 - 1899)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory




Founders of DG

Felix Klein (1849 - 1925)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory




Founders of DG

Gregorio Ricci-Curbastro (1853 - 1925)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory




Founders of DG

Tullio Levi-Civita (1873 - 1941)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory




Founders of DG

Elie Cartan (1869 - 1951)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory




Founders of DG

Henri Cartan (1904 - 2008)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory




Founders of DG

Charles Ehresmann (1905 - 1979)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory




Founders of DG

Hassler Whitney (1907 - 1989)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory




Founders of DG

Jean-Louis Koszul (1921 - )


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Electromagnetic StoryStarring


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory


Luigi Galvani (1737-1798)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory


Alessandro Volta (1745-1827)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory


Gian Domenico Romagnosi (1761-1835)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory


Hans Christian Ørsted (1777-1851)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory


Andre-Marie Ampere (1775 - 1836)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory


Johann Carl Friedrich Gauss (1777 - 1855)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory


Francesco Zantedeschi (1797-1873)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory


Joseph Henry (1797 - 1878)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory


Michael Faraday (1791 - 1867)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory


James Clerk-Maxwell (1831 - 1879)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory


Joseph John Thomson (1856 - 1940)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory



elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Vector fields in Electromagnetics


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory


Josiah Willard Gibbs (1839 - 1903)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory


Oliver Heaviside (1850 - 1925)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory


Heinrich Rudolf Hertz (1857 - 1894)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Zero spatial velocity∮∂Σin

g · E = −∫

Σin

µ · (∂θ=0 B) Henry− Faraday(1831)∮∂Σout

µ · B = 0 Gauss(1831)

∮∂Σout

g ·H =

∫Σout

µ · (∂θ=0 D + J) Ampere(1826)

Maxwell(1861)∮∂Σout

µ ·D =

∫Σout

ρµ Gauss(1835)

with Σout a bounded connected surface and Σout boundedconnected domain in S . Applying Ampere law to closedsurfaces Σout = ∂Ω , we get

∂θ=0 ρ+ div J = 0

the equation of continuity.


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Relativity Story

Length contraction and time dilation effects

Woldemar Voigt (1850 - 1919)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Relativity Story


George Francis FitzGerald (1851 - 1901)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Relativity Story


Hendrik Antoon Lorentz (1853 - 1928)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Relativity Story


Albert Einstein (1879 - 1955)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Relativity Story

Space-time metric

Henri Poincare (1854 - 1912)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Relativity Story

Space-time metric

Hermann Minkowski (1864 - 1909)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Relativity Story

General relativity

Christian Felix Klein (1849 - 1925)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Relativity Story

General relativity

David Hilbert (1862 - 1943)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Albert Einstein (1905)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

ON THE ELECTRODYNAMICS

OF MOVING BODIES

By A. EINSTEIN

June 30, 1905

It is known that Maxwell’s electrodynamics—as usually understood at thepresent time—when applied to moving bodies, leads to asymmetries which donot appear to be inherent in the phenomena. Take, for example, the recipro-cal electrodynamic action of a magnet and a conductor. The observable phe-nomenon here depends only on the relative motion of the conductor and themagnet, whereas the customary view draws a sharp distinction between the twocases in which either the one or the other of these bodies is in motion. For if themagnet is in motion and the conductor at rest, there arises in the neighbour-hood of the magnet an electric field with a certain definite energy, producinga current at the places where parts of the conductor are situated. But if themagnet is stationary and the conductor in motion, no electric field arises in theneighbourhood of the magnet. In the conductor, however, we find an electro-motive force, to which in itself there is no corresponding energy, but which givesrise—assuming equality of relative motion in the two cases discussed—to elec-tric currents of the same path and intensity as those produced by the electricforces in the former case.

Examples of this sort, together with the unsuccessful attempts to discoverany motion of the earth relatively to the “light medium,” suggest that thephenomena of electrodynamics as well as of mechanics possess no propertiescorresponding to the idea of absolute rest. They suggest rather that, as hasalready been shown to the first order of small quantities, the same laws ofelectrodynamics and optics will be valid for all frames of reference for which theequations of mechanics hold good.1 We will raise this conjecture (the purportof which will hereafter be called the “Principle of Relativity”) to the statusof a postulate, and also introduce another postulate, which is only apparentlyirreconcilable with the former, namely, that light is always propagated in emptyspace with a definite velocity c which is independent of the state of motion of theemitting body. These two postulates suffice for the attainment of a simple andconsistent theory of the electrodynamics of moving bodies based on Maxwell’stheory for stationary bodies. The introduction of a “luminiferous ether” will

1The preceding memoir by Lorentz was not at this time known to the author.

1

AlbertEinstein (1905)


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Maxwell-Hertz equations - (Einstein 1905)

1c∂X∂t = ∂N

∂y −∂M∂z

1c∂L∂t = ∂Y

∂z −∂Z∂y

1c∂Y∂t = ∂L

∂z −∂N∂x

1c∂M∂t = ∂Z

∂x −∂X∂z

1c∂Z∂t = ∂M

∂x −∂L∂y

1c∂N∂t = ∂X

∂y −∂Y∂x


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Differential forms vs. vector fields

ω1E = g · E electric field

ω2B = ω3

µ · B magnetic vortex

ω1H = g ·H magnetic field

ω2D = ω3

µ ·D electric flux

ω1B = g · A magnetic potential

ω2J = ω3

µ · J electric current

ω3ρ = ρω3

µ electric charge

ω0E = VE electric potential


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Space-time split

Z field of time-arrowsR = dt ⊗ Z projector on time-arrowsP = I− R projector on space slices


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Space-time theory of electromagnetics

ω2B = i↓Ω2

B ∈ Λ2(VS ;R) magnetic vortex

ω1E = i↓Ω1

E ∈ Λ1(VS ;R) electric field

with i spatial immersion.

Ω2B := P↓Ω2

F

−Ω1E := P↓(Ω2

F · V)

P projector on the spatial slicesV = Z + PV space-time velocityv = i↓(PV) spatial velocity.Representation formula

Ω2F = Ω2

B − dt ∧ (Ω1E + Ω2

B · V) .


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Faraday law in space-time

Closedness of Faraday two-form in the trajectory manifold isequivalent to the spatial Gauss law for the magnetic vortex andto the spatial Henry-Faraday induction law, i.e.

d Ω2F = 0 ⇐⇒

dS ω2B = 0

LSV ω2B + dS ω

1E = 0

with the integral formulation

∂θ=0

∫ϕSθ (Σin)

ω2B = −

∮∂Σin

ω1E

for any inner-oriented surface Σin in a spatial slice.


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Faraday law

−ω1E = LSV ω1

B + dS ω0E

= LSZ ω1B + Lv ω

1B + dS ω

0E

= LSZ ω1B + (dS ω

1B) · v + dS (ω1

B · v) + dS ω0E

= LSZ ω1B + ω2

B · v + dS (ω1B · v) + dS ω

0E

Usual formulation with dS (ω1B · v) dropped

−ω1E = LSZ ω1

B + ω2B · v + dS ω

0E


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Synoptic table ( v = 0 )

new old

(E‖ ,E⊥) → (γ E‖ ,E⊥) (E‖ , γ (E⊥ + w × B))

(B‖ ,B⊥) → (B‖ , γ (B⊥ − (w/c2)× E)) idem

(H‖ ,H⊥) → (γH‖ ,H⊥) (H‖ , γ (H⊥ −w ×D))

(D‖ ,D⊥) → (D‖ , γ (D⊥ + (w/c2)×H)) idem

(J‖ , J⊥) → (J‖ , γ J⊥) (γ (J‖ − ρw) , J⊥)

ρ → γ (ρ− g(w/c2 , J)) idem

VE → VE γ (VE − g(w,A))

(A‖ ,A⊥) → (γ (A‖ − (w/c2)VE) ,A⊥) idem


elettromagnetica

Giovanni Romano

Electromagnetics


Simplifications


New theory

Denoting by ∇ the Euclid connection in spatial slices andassuming that the observer measures1) a magnetic potential independent of time,

LSZ ω1B = 0

2) a spatially constant scalar electric potential,

dS ω0E = 0

3) a spatially constant magnetic vortex field,

∇ω2B = 0

the formula for the electric field may be evaluated to get

−ω1E = (dS ω

1B) · v + dS (ω1

B · v) = ω2B · v−

1

2ω2

B · v =1

2ω2

B · v ,

or in terms of vector fields E = 12 v × B which is just one-half of

what is improperly called the Lorentz force.

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