Black Holes and Curved Spacetime

1. Black Holes and Curved Spacetime

2. Outline • Gravity • Action at a Distance • Principle of Equivalence • Spacetime • Black Holes • Static (Schwarzchild) • Rotating (Kerr) • Evidence for Black Holes

3. Gravity: Action at a distance • Newton (1642-1727) • “It is inconceivable, that inanimate brute matter should, without the mediation of something else, …, operate upon, and affect other matter without mutual contact;…And this is one reason, why I desired you would not ascribe innate gravity to me.” • “That gravity should be innate, inherent, and essential to matter, so that one body may act upon another, at a distance through vacuum, without the mediation of anything else…, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking, can ever fall into it.”

4. Gravity: Mass • Two kinds of mass in Newton’s physics: • Inertial mass, which measures resistance to forces (F = ma) • Gravitational mass, which measures the strength of gravitational force between two objects (F = Gm1m2/r2) • For Newton there was no physical reason why inertial and gravitational masses should be related.

5. Gravity: Acceleration • Of Elephants and Students • Elephants and students, dropped from the top of a circus tent, and indeed all objects, fall towards the Earth at the same rate, a fact first noted by Galileo (1564-1642). • This experimental fact follows if the inertial and gravitational masses were equal to each other, a result that was first noted by Newton. • But it was a great puzzle to Newton why two apparently unrelated physical quantities, namely, inertial and gravitational mass, should have the same value.

6. Gravity: The Principle of Equivalence • Albert Einstein (1879-1955) • In 1907 he had a pivotal insight: A falling observer does not experience gravity…He described this as the“happiest thought of my life.” • This insight led Einstein to propose the principle of equivalence: • A local frame of reference in free fall in a gravity field is indistinguishable from a local frame of reference far from a gravity field. • Einstein explained this with his famous elevator gedanken (thought) experiment.

8. free space The Principle of Equivalence: free space view same as free fall view Light ray free fall inside view free fall outside view

9. Gravity as Geometry • In 1912, Einstein had another profound insight: • If all accelerated frames of reference were equivalent then Euclidean geometry cannot hold in all of them. That is, the geometry of space is not necessarily Euclidean. • Einstein did not know the mathematics he needed to turn his idea into a physical theory so he turned to his friend Marcel Grossmann who directed Einstein to the works of Riemann, Ricci and Levi-Civita on differential geometry. • “…in all my life I have not labored nearly so hard, and I have become imbued with great respect for mathematics..” Albert Einstein

10. Gravity as geometry (contd.) • In 1915, Einstein published his General Theory of Relativity, which provided a radical explanation of free fall motion and therefore of gravity. • The free fall motion of objects depends on the geometry of the space through which they move. • Objects in free fall move the same way simply because they experience the same spatial geometry • Moreover, the geometry of space and time is determined by matter and energy.

11. The future t r x O y now The past time Spacetime The light cone: the set of all paths that would be traveled by light emitted from an event. Worldlines of two light rays on light cone. ds2 = dt2 - dl2

12. Black Holes • In 1783 John Michell pointed out the possibility that a sufficiently massive and compact star could have an escape velocity greater than that of light. If so, light would be unable to escape from such a star. • In 1916, just months after Einstein published his theory of general relativity, Karl Schwarzchild, while a frontline soldier in World War I, found the first solution to Einstein’s equation. • Schwarzchild’s solution was later understood to describe the spacetime geometry around what, in 1969, John Wheeler coined a black hole.

13. The Schwarzchild Radius Black Hole: The Event Horizon • A black hole is a region of curved spacetime, also called the Schwarzchild sphere, from which light cannot escape. • The “surface” of the black hole is called an event horizon. It is a horizon in the sense that events within the black hole can never be seen from outside.

14. Black Hole: Static (Schwarzschild) Event horizon (Static limit – below this limit it is impossible to remain stationary) Photon sphere 3/2 rS rS Within the black hole the spacetime is dynamic and has the topology of a sphere times a line Singularity This is not so much a place as the time in your future when time stops

15. Flying over the Photon Sphere

16. Black Hole: Rotating (Kerr) Static limit Outer event horizon Outer event horizon Inner event horizon Ergosphere Ring singularity From Black Holes - A Traveller's Guide, by Clifford Pickover (Wiley 1996).

17. Evidence for Black Holes

18. M87 Distance: 50 Mly Length of jets: 5000 ly Black hole: ~3 billion Msun