From An Apple To Black Holes Gravity in General Relativity

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1 From An Apple To Black Holes Gravity in General Relativity

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3 Gravity as Geometry Central Idea of General Relativity Gravitational field vs magnetic field Uniqueness of trajectory in space and time Uniqueness explained in terms of geometry Four-dimensional union of space and time: spacetime Mass curves the geometry of nearby spacetime Straightest possible path in curved spacetime

4 Gravity as Geometry How does that work? Imagine the earth (Sphere) Why do airplanes not fly in straight lines? Shortest distance on a sphere: great circles

5 Gravity as Geometry What does that mean? How can we describe geometry? Use of differential and integral calculus Reduce all geometry to specification of distance The most general geometry is specified by the distances between nearby points (!) Distances along curves by integration Areas, volumes etc. by multiple integrals

6 Gravity as Geometry How to get a geometry? The geometry of a sphere use spherical coordinates calculate the basis vectors x=r (sin θcosφ,sin θsinφ,cosθ) calculate all inner products for each basis vectors with each one g ij = e i e j you get a 3x3 matrix (known as metric tensor) with following elements: g ij =( θ) 0 r r 2 sin

7 Gravity as Geometry What is now the most general specification of nearby points? It's the line element! The line element squared can easily be calculated with our matrix we got before: ds 2 =g ij dx i dx j Because of that we get for our sphere: ds 2 =dr 2 +r 2 dθ 2 +r 2 sin θd Φ 2

8 Gravity as Geometry Four dimensional newtonian line element ds 2 = (1+ 2Φ where with the field equation c 2 )(c dt)2 +(1 2Φ Φ(r)= G M r So far for the geometric newtonian c 2 )(dx2 +dy 2 +dz 2 ) Δ Φ=4 πgμ What happens now in General Relativity?

9 Einstein's Equation The presence of matter produces curvature How does the equation look? The schematic form can be written down as a measure of local spacetime curvature = a measure of matter energy density Einstein's equation It relates the spacetime curvature to its source: the mass-energy of matter Maxwell's field equations as analogy

10 Einstein's Equation Ten second-order partial differential equations for the metric coefficients Difference to Maxwell's equations: nonlinear! How does the mathematic form of the equation look? R ik 1 2 g ik R+Λ g ik =8 π G c 4 T ik Ricci tensor Metric Cosmological Trace of the Ricci constant tensor Energy-momentum tensor

11 Einstein's Equation Ricci tensor: Expresses the curvature of Riemannian manifolds Cosmological constant: Value of the energy density of the vacuum of space in other words: It is the simplest possible form of Dark Energy as it is constant in space and time Energy-momentum tensor: Describes the density and flux of energy and momentum in space Generalization of the newtonian stress tensor

12 Einstein's Equation For what do we solve the equation? To get the metric coefficients! That means that the metric tensor solves the equation Why is this interesting? Imagine the simplest form of the Einstein equation...

13 Einstein's Equation Simplest curved spacetime: The one with the most symmetry! Imagine a spherical star We want to describe the geometry of empty space of such a star What does empty space mean? Vacuum! Forget the cosmological constant and the energy-momentum tensor

14 Einstein's Equation That yields the vacuum Einstein equation: R ik =0 Remember that that is the relativistic generalization of Δ Φ=0 Now we have 10 second-order partial differential equations for the ten metric coefficients

15 Einstein's Equation Why does he say for the metric coefficients when the equation is R ik =0? The Ricci tensor is defined by R αβ = Γ γ α β Γ γ α γ x γ x β +Γ γ α β Γ δ γδ Γ γ δ αδ Γ β γ Where the gammas are called Christoffel symbols The Christoffel symbol on the other hand is defined by the metric Γ β γ α = 1 2 gαδ ( g δβ x γ + g δ γ x β g β γ x δ )

16 Einstein's Equation This set of equations already show that even for vacuum this is already a lengthy calculation To save time I will only mention 3 solutions of many more The Minkowski metric (flat space) The Schwarzschild metric (outside of a star in vacuum) The Kerr metric (rotating black hole)

17 Schwarzschild Geometry After a lengthy calculation we got the Schwarzschild metric as a possible solution to the vacuum Einstein equation The metric is defined by (c,g=1 units) ds 2 = (1 2M r )dt 2 +(1 2M 1 r ) dr 2 +r 2 d Ω 2 where dω 2 =d θ 2 +sin 2 θd Φ 2

18 Schwarzschild Geometry But what happens at r = 0 and r = 2M? Good question! Is there maybe a transformation that we can avoid the singularities? There is one for r = 2M: t=v r 2 M log r 2M 1 Therefore we get the Eddington-Finkelstein metric ds 2 = (1 2M r )dv 2 +2 dv dr+r 2 d Ω 2

19 Schwarzschild Geometry The singularity at r = 0 stays The singularity at r = 2M was just a coordinate singularity Are there objects in the universe that have a singularity at r=0? Maybe if the gravitational collapse never stops In theory such objects are called Black Holes

20 Black Holes What are black holes? Imagine a star that cannot perform nuclear fusion anymore That means that there is no gas pressure countering the gravitational force Mass of star under 1.4 solar masses electrons counter the gravitational collapse Pauli exclusion principle The core will contract to a white dwarf which has a radius of a few thousand kms

21 Black Holes Mass between 1.4 and 3 solar masses electrons get pulled into the atom nuclei that transforms the protons into neutrons the neutrons now counter the gravitational collapse Pauli exclusion principle The core of star now is called a neutron star with a radius of a few kms

22 Black Holes What happens for masses over 3 solar masses? Black holes! The neutrons cannot prevent the gravitational collapse anymore The core of the star has transformed into a black hole The black hole seriously disturbs the spacetime around it The whole mass of the core is now within the Schwarzschild radius aka the r = 2M coordinate singularity Our sun has a Schwarzschild radius of about 3 km r= 2G M c 2

23 Black Holes The Schwarzschild radius is better known as the event horizon Boundary in spacetime beyond which events cannot affect an outside observer In other words: It is the point of no return The point at which the gravitational pull becomes so great as to make escape impossible, even for light

24 Black Holes Object approaching the horizon from observer's side appears to slow down and never quite passes through the horizon It's image is becoming more and more redshifted as time elapses The object, however, does in fact pass the horizon in a finite amount of proper time For a black hole with 30 solar masses it will take about 0.2 milliseconds to get from the horizon to the singularity

25 Black Holes But what does that even have to do with General Relativity? Well all the descriptions and theories arose with General Relativity Anything you heard/saw here is a direct consequence of the Einstein equation Have there been direct observations of black holes yet? No! But indirect, for example the detection of gravitational waves from a binary black hole merger was an indirect observation of a binary black hole system

26 A star walks into a black hole but it doesn't seem phased. The Black hole turns to the star and says, Sir, I don't think you understand the gravity of this situation.

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