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Astroparticle Physics Astroparticle Physics Luca Doria Institut für Kernphysik Johannes-Gutenberg Universität Mainz Lecture 2 Brief History of GR 1907: While working at the Bern’s Patent Office, A. Einstein realized how to start generalizing special relativity to generic reference frames. 1908: First paper about acceleration and relativity by A. Einstein. In this paper he states the Equivalence Principle and derives time dilation caused by gravitational fields. 1911: Second paper by A. Einstein on time dilation by gravitational fields where also light deviation by massive bodies was approximately derived. 1912: Einstein consults with M. Grossman about non-euclidean geometry. October 1915: First guess: Rij = Tij November 1915: Einstein publishes the General Theory of Relativity as we know it today. D. Hilbert obtained the same equations almost at the same time. 1919: Eddington confirms the deviation of light formula from GR using a solar eclipse in Brazil. 1959: Pound-Rebka Experiment (gravitational red shift) 1971: Hafele-Keating Experiment (time dilation) 1974: Hulse-Taylor binary pulsar. 2004: Gravity Probe-B and frame dragging (published in 2011) 2016: Direct detection of gravitational waves by LIGO Sommersemester 2018 Luca Doria, JGU Mainz 2 The Equivalence Principle Observer in an uniform gravitational field Accelerated Observer How do you tell the difference? Sommersemester 2018 Luca Doria, JGU Mainz 3 The Equivalence Principle What about this case? Sommersemester 2018 Luca Doria, JGU Mainz 4 The Equivalence Principle Strong Equivalence Principle: At every space-time point in a gravitational field, it is possible to choose a locally inertial coordinate system such that in a sufficiently small neighbourhood of that point, the laws of nature can be expressed in the same form as in an unaccelerated coordinate system. Weak Equivalence Principle: Change “laws of nature” with “laws of motion of free-falling bodies” (gravity). Sommersemester 2018 Luca Doria, JGU Mainz 5 CHAPTER 3. GENERAL RELATIVITY bodies. Clearly the strong version implies the weak on but not vice-versa. CHAPTER 3. GENERAL RELATIVITY bodies. Clearly the strong version implies the weak on but not vice-versa. 3.2 Free-FallingCHAPTER 3. GENERAL Bodies RELATIVITY bodies. Clearly the strongLet’s3.2 version tryFree-Falling implies to translate the weak in on Bodies mathematical but not vice-versa. formulas the ideas contained in the EP. Consider a free-falling body: according to the EP, there must exist lo- Let’s try to translate in mathematical formulas the ideas contained in the 3.2 Free-Fallingcally Bodies a coordinate system where the effect of gravitation vanishes (inside Einstein’sEP. Consider elevator) a free-falling and body: the equation according of to motion the EP, there in flat must space exist lo-time is cally a coordinate system where the effect of gravitation vanishes (inside Let’s try to translate in mathematical formulas the ideas contained in the Einstein’s elevator) and the equationd2 ofxµ motion in flat space time is EP. Consider a free-falling body: according to the EP, there must exist= lo-0 , (3.1) cally a coordinate system where the effect of gravitation vanishesd2xµ dt2 (insideFree-Falling Bodies = Einstein’s elevator) and the equation of motion in flat space2 time0 is , (3.1) µ n dt where dt = hµndx dx is the proper time and hµn the Lorentz metric. 2 µ µ n whered xdt = −hµndx dx is the proper time and hµn the Lorentz metric. µ Let’s get out= 0− from , theEquation elevator of motion and change (3.1)for a free-falling to new body coordinatesµ x , which canLet’s bed gett whatever2 out from we the elevator want (a and curvilinear change to system, new coordinates an acceleratedx , which or rotating can be whatever we want (a curvilinear system, an accelerated or rotating where dt = h dxµdxn is the proper time and h the Lorentz metric. − µn system,system, etc..) etc..)Changing toµ na generic coordinate system: Let’s get out from the elevator and change to new coordinates xµ, which can be whatever we want (a curvilineardd system,∂x∂xa dxa anµdx acceleratedµ ∂xa d∂x2xa orµ d rotating2xµ∂2xa dx∂2µxdxa n dxµ dxn = = + + = 0 .= 0 (3.2) . (3.2) system, etc..) dt ∂xµ dµt ∂xµ dt2µ ∂2xµ∂xn dtµ dtn dt✓ ∂x d◆t ∂x dt ∂x ∂x dt dt a µ a 2 µ ✓2 a µ ◆n d ∂x dx Multiplying∂x d x the∂ lastx equationdx dx by ∂xl/∂xa and recognizing the presence µ = µ 2 + µ n = 0 .l (3.2) a dt ∂x dt Multiplyingof the∂x Christoffeldt the∂x symbol∂ lastx dt equationdt by ∂x /∂x and recognizing the presence ✓ ◆ Equation of motion: of the Christoffell a symbol Multiplying the last equation by ∂x /∂x and recognizing2 l theµ presencen d x l dx dx of the Christoffel symbol 2 2+lGµn µ= 0n . (3.3) dt d x dtl ddxt dx + Gµn = 0 . (3.3) d2Thexl last resultdxµ dxn is quite interesting:dt2 the presencedt dt of the gravitational field + Gl = 0 . (3.3) dtcan2 beµ seenn dt asd at curvature of space-time where free-falling particles follow The last result is quite interesting: the presence of the gravitational field The last result is quite interesting:a geodetic the in presence that space. of From the gravitational the coordinate field transformation formula and candt = be seenh dx asµdx an curvature, it is clear that of space-time the metric tensor where of thefree-falling curved space particles is follow can be seen as a curvature of space-time− µn where free-falling particles follow a geodetic in that space. Fromarelated geodeticSommersemester the to coordinate the2018 in Lorentzian that transformation space. flat From space formula theLuca by Doria, coordinate JGU and Mainz transformation formula6 and µ n dt = h dxµdxn, it is cleardt = that thehµn metricdx dx tensor, it is of clear the curved that the space metric is tensor of the curved space is − µn − ∂xa ∂xb related to the Lorentzianrelated flat space to by the Lorentziangµn = flat space= byh . (3.4) ∂xµ ∂xn ab ∂xa ∂xb a b gµn = = hab .∂x ∂x (3.4) ∂xµ ∂xn gµn = = h . (3.4) 3.3 Non-Relativistic Limit∂xµ ∂xn ab 3.3 Non-RelativisticSo far Limit our calculations were relativistic and now we would like to see if 3.3what weNon-Relativistic derived can be reduced to the Limit know non-relativistic result, which So far our calculations were relativistic and now we would like to see if 26 what we derived can be reducedSo far to our the calculations know non-relativistic were relativistic result, which and now we would like to see if what we derived26 can be reduced to the know non-relativistic result, which 26 CHAPTERCHAPTER 2. 2. GEOMETRICAL TOOLS TOOLS CHAPTER 2. GEOMETRICAL TOOLS AA general general mixed mixed tensor has has the the following following covariant covariant derivative: derivative: a1..ar A general mixed tensor has the following covariant derivative: a1..ar ∂ ec T a1..ar ∂ ec Tbb..1b..bs ( e T)a1..ar = rr 1 s (2.37) (a ..eacT) b1..br = c (2.37) ∂ T rr1 cr b1..br ∂∂xxc a ..a ec b ..bs 1 r a 1da ..a a a1..ar 1d ( e T) = r a11 da22..arr ar r a1..ar (2.37)1d c b1..br ++GGdcTTc b ..b ++......++GGdcTTb ..b−− (2.38)(2.38) r ∂dcx b11..bss dc b1..1bs s a1 da2..ar ar da1..ara11..dar d a1..ar +G T + ... + G GTd Ta−1..ar ... Gd Ta1..ar (2.38) (2.39) dc b1..bs dcGbb1c..Tbsdb ..b ... G bscT b ..b d (2.39) b11c db22..bss bsc b1..1bs s 1d1 − −− −− −− Gd Ta1..ar ... Gd Ta1..ar (2.39) Notice− b1c db that2..bs the− derivation− bsc b1..bs of1d covariant indices brings a negative sign in CHAPTER 2.Notice GEOMETRICAL that the derivation TOOLS of− covariant indices brings a negative sign in Notice that the derivationfrontfront of of ofGG.. covariant indices brings a negative sign in front of G. A general mixed tensor has the following covariant derivative: 2.102.10 CalculationCalculation of of the thea1..ar Christoffel Christoffel Symbols Symbols ∂ e T CHAPTERa1..ar 3.c b1 GENERAL..bs RELATIVITY 2.10 Calculation of the( e ChristoffelT) = r Symbols (2.37) HavingHaving a a procedure procedurer c b for for1..br calculating calculating∂xc derivatives derivatives of of tensors tensors along along a a direction direction a1 da2..ar ar a1..ar 1d Having a proceduregivengiven for calculating by by+ a aG tangent tangentT derivatives vector,+ ... + weG we of tensorsT need need− now now along a a way a way direction to to calculate calculate(2.38) the the Christoffel Christoffel bodies. Clearly the strong versiondc b1..bs implies thedc b weak1..bs on but not vice-versa. given by a tangentsymbols. vector,symbols. weG needd Ta1 now..ar a... wayG tod T calculatea1..ar the Christoffel (2.39) b1c db2..bs bsc b1..bs 1d symbols. First,First, we we− have have to know− the the− following following− property property First,3.2 weNoticeFree-Falling have to that know the the derivation following Bodies of property covariant indices brings a negative sign in front of G. rrggµµnn==00 . (2.40) (2.40) g = 0 .rr (2.40) Let’s try to translate in mathematicalrr µn formulas the ideas contained in the TheThe covariant covariant derivative of of the the metric metric tensor tensor is is zero. zero. This This property property is is EP. Consider a free-falling body: according to the EP, there must exist lo- The covariant derivativecalledcalled metric ofmetric the compatibility compatibility metric tensorandand is zero.
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