Einstein s route to new description of gravity 1)

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1 Einstein s route to new description of gravity 1) ( probably the most beautiful of existing physical theories ) Vladimír Balek Department of Theoretical Physics, Comenius University Stará Lesná, September 19, ) dedicated to the Founding Father of the School in Stará Lesná Pavel Bóna

2 L. D. Landau, E. M. Lifshitz: Teoria polya (Field Theory), 6th ed., Moscow, Nauka (1973) p. 291

3 why principle of relativity? It is known that Maxwell s electrodynamics as usually understood at the present time when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. E., On the electrodynamics of moving bodies (1905) a theory with v rel only? NO, just a theory in which I = I ind

4 equivalence of S in is nice, but... in the theory there remains absolute space up to uniform motion, but still; Mach: WRONG, the theory should contain r rel, v rel only S in, m in should be given by the action of distant stars E. (1916, but surely thought of it before): 2 rotating fluid spheres in empty space should behave identically S rot, too, should be equivalent; GR: spheres rotate with respect to M asym, which is determined by distant matter (= M app in an expanding universe)

5 gravity enters the stage E. s happiest thought : m in = m grav, see Leaning Tower in Pisa and cannon balls F in = F grav principle of equivalence: accelerated frame ( antilift ) is equivalent to frame in rest in g E.: REASON of m in = m grav ; The Meaning of Relativity (1921): It is, however, clear that science is fully justified in assigning such a numerical equality only after this numerical equality is reduced to an equality of the real nature of the two concepts.

6 slow time, quick time The heuristic value of this assumption (PE) rests on the fact that it permits replacement of a homogeneous gravitational field by a uniformly accelerated reference frame, the latter case being to some extent accessible to theoretical treatment. light from floor to ceiling: v DOWN = 0 hills A a B start from the same point & v UP > 0 the hill B travels a longer path than the hill A t UP > t DOWN, time flows downside slower than upside; PE: ditto in gravitational field ( 11 km per day in GPS!)

7 solution will be in the Gauss theory of surfaces dilation of t variable c (dl = cdt id = cdt, c UP > c DOWN ) variable (00)-component of metric tensor (ds 2 = c 2 dt 2 dx 2 ) COMPLETE metric tensor (ds 2 = g µν dx µ dx ν ) dilation s h in the same way as stretching of parallels s θ, THEREFORE spacetime is curved in the direction of h otherwise it is antilift ( a plane with coordinates (r, φ))

8 equations from Outline E. Grossmann: Outline of a Generalized Theory of Relativity and of a Theory of Gravitation (1913): G µν 0 = ψ µνσ,σ, ψ µνσ = g µν,σ : T µ ν ;ν! = t µ ν,ν t µν ψ 2 & G µν 1 = κ E t µν + lowering term non-covariance? hole argument : outside matter (in a hole ) a covariant theory has solutions g µν (x) as well as g µν (x ) it is not unique? NO, x for the same g µν x

9 four Thursdays E. s lectures at Prussian Academy of Science in November 1915: Thursday 4th equations with reduced covariance : R (1) µν = κ E T µν, R (1) µν = half of R µν (= Γ κ µν,κ Γ λ µκγ κ λν, reduces to R µν for g = const) + condition on x µ : [g µν (ln g),µ ],ν = κ E T covariant equations with T = 0 Thursday 11th Thursday 18th precession of the perihelion of Mercury Thursday 25th covariant equations with arbitrary T : R µν (1)! = κ E (T µν 1 2 Tg µν) [g µν (ln g),µ ],ν = 0 R µν = κ E (T µν 1 2 Tg µν)

10 I want to know how God created this world. I m not interested in this or that phenomenon, in the spectrum of this or that element. I want to know His thoughts, the rest are details. Albert Einstein