Curved Spacetime III Einstein's field equations Dr. Naylor Note that in this lecture we will work in SI units: namely c 1
Last Week s class: Curved spacetime II Riemann curvature tensor: This is a tensor Properties of the Riemann tensor? Double covariant derivative on a vector field V, is equivalent to parallel transport around a closed curve: Geodesic deviation between two geodesics, where ξ determines the distance between the two geodesics, Let s continue with part III 2
The Bianchi identities and Ricci tensor By using a locally inertial frame, LIF () ;!, ) then differentiation the Riemann tensor leads to the Bianchi identities: Ricci tensor comes from contracting the Riemann tensor: The Ricci scalar is defined by contracting the Ricci tensor: Exercise: Show that Ricci contracting the Bianchi identities leads to: Hint: Use g αβ;µ =g αβ ;µ =0 and use anti-symmetry in Riemann tensor 3
Einstein tensor Exercise: Show that contracting a second time gives and hence that where we have defined G µν is called the Einstein tensor, because it was Einstein who first realized about its importance in general relativity. As we shall see in the rest of this lecture the Einstein gravitational field equations are given by where κ is a constant of proportionality and T µν is the stress energy tensor for matter, which we discussed in a previous lecture This equation tells us how matter curves spacetime and also how space tells matter to move (Wheeler) Note that the Bianchi identities imply that T αβ ;β =0 4
On the previous slide we gave the equation Einstein field equations we shall now show that κ = 8π G/ c 4 where c is the speed of light and G is Newton s gravitational constant. We need to know how 1. particles move in curved space (Φ c 2 ) F = -mrφ ) 2. matter curves spacetime Let s first answer question 2, using the gravitational potential (Poisson s) formula which relates the gravitational field Φ to the mass density ρ When in a vacuum ρ = 0 consider the acceleration of two neighbouring particles at x and x+ξ ) 5
Their separation therefore corresponds to where we have used This is very similar to the geodesic deviation equation: Both equation have two indices (however, note that Newtonian i=1,2,3 and Minkowski α=0,1,2,3 ) Newtonian vacuum equation is r 2 Φ = 0 ) Thus, we can infer that and since V is arbitrary we have that R µν = 0 These are hence the vacuum Einstein field equations In non-vacuum (ρ 0) the GR version of must contain the stress energy tensor T µν 6
Non-vacuum equations For example consider the stress-energy for a perfect fluid: T αβ = (ρ + p/c 2 ) U α U β + p g αβ and T αβ,α =0 However via the equivalence principle we take,! ; and thus ) T αβ ;α =0 Thus the full-non vacuum equations should be of the form where O is a second order differential operator which is a and κ is a constant When T=0 simplest choice will be But given that T αβ ;α = 0 then we must also have Then from the double contracted Bianchi identities 7
Cosmological constant Thus the Einstein field equations must be However we can also add a cosmological constant term Λ In vacuum T αβ = 0, we can take the trace of the field equations ) Exercise: Show the above equation leads to R αβ = Λ g αβ Hint: use the fact that trace of the metric is g α α = 4 If Λ=0 we get the vacuum field equations, thus, in this context Λ is sometimes called vacuum energy density R αβ is symmetric so we have 10 equations for the 10 metric components g αβ, but there are 4 degrees of freedom in x α coordinates. Thus, 6 metric components a determinable and this comes from the constraint G αβ ;α = 0 8
Einstein s greatest blunder/mistake?!! When Einstein first wrote down his field equations he decided to drop his cosmological constant term Λ Why? Because he favored a static universe and Λ > 0 implied the universe was expanding However, in 1927 Hubble showed conclusively that all galaxies are moving away from each other: The universe is expanding!!! Hence in hindsight, Einstein is quoted as saying that dropping Λ was his biggest blunder, because of he had included it in his equations he could have predicted the expansion of the universe a decade ahead of observation! In recent cosmology too Λ is playing a very important role: THE UNIVERSE IS NOT SLOWING DOWN 9
Newtonian limit: weak field approximation Now we will return to Question 1, which is how particles move in curved space Assume that free particles follow timelike or null geodesics Furthermore assume that v/c 1 and chose a coordinate system which is locally Minkowski (equivalence principle) and assume Here ε 1 and we neglect order ε 2 and higher Exercise: Given that g αβ g αβ = δ α β show that the metric inverse is Show that the Christoffel symbol in this limit becomes 10
Geodesics in Newtonian limit In terms of proper timeτ for a freely falling object we have in the 2 nd step we have used that fact the t ¼ τ for v c Given that dx i /dt =O(v) we neglect (Γ γ ij dxi /dt dx j dt ) ) where we have used the fact that dx 0 /dt = c Then because Note in the second step we neglected time derivatives Thus, the spatial geodesic equation becomes 11
However, by Newtonian theory We have and therefore if we identify Then we find the weak field metric to be Now let s return to the field equations Exercise: show that taking the trace gives Using this trick we can write the field equations as Exercise: Taking the trace of T αβ = (ρ + p/c 2 ) U α U β + p g αβ show that and hence that 12
In the non-relativistic limit We have ρ À p/c 2 and hence Exercise: show that the 00 component is R 00 = ½ κρc 2 to 1 st order in ε Now because we are working in a LIF we have but assuming spatial derivatives dominate implies However Poisson s field equation implies that the constant κ = 8π G/ c 4 and thus, This was the main GOAL of this course! 13
Summary of links between SR and GR Taken from Peter Dunsby s lecture notes 14
Next time: Black Holes We will investigate the Schwarzschild metric And consider the time dilation effects Particle orbits Special coordinates The event horizon 15
Final year exam Tuesday, January 17 th, last class 60 minutes 30 % of the mark You must get more than 50% (1/2 marks) in the test! To pass the course 16