GENERAL RELATIVITY CURVED SPACE by Robert Nemiroff Michigan

GENERAL RELATIVITY & CURVED SPACE by Robert Nemiroff Michigan Tech

Physics X: About This Course • Officially "Extraordinary Concepts in Physics" • Being taught for credit at Michigan Tech o Light on math, heavy on concepts o Anyone anywhere is welcome • No textbook required o Wikipedia, web links, and lectures only o Find all the lectures with Google at: § "Starship Asterisk" then "Physics X" o http: //bb. nightskylive. net/asterisk/viewforum. php? f=39

General Relativity: Overview • GR: Gravity is not a force o It is a non-Euclidean geometry o ". . . there is no theory of gravity beyond the mathematical form. " - Richard Feynman • Very close to Newtonian gravity • Tests o precession of Mercury o deflection of starlight • Mechanism: Actually -- none!

GENERAL RELATIVITY: HISTORY • • Created by Einstein between 1905 and 1915. Published incorrect theory of gravity in 1912. Test of incorrect theory clouded out. Published GR in 1915. Key GR test passed in 1919. Remains our most accurate theory of gravity. Many scientists feel that: SR would have developed without Einstein, although Einstein got there first, but GR would not have been developed without Einstein.

General Relativity: Principles and Theorems • "Local" effects are different than "global" effects o Local = very nearby • Space near mass "curves" • Time near mass "slows" • Equivalence Principle o All things fall together • Birkhoff's Theorem o Ignore symmetric outside stuff • Gravitomagnetism o is to G as B is to E.

General Relativity: Concept of Spacetime • Spacetime o Space and time combined into one coordinate system o time like a fourth spatial dimension o still three spatial dimensions o lines represent coordinate system

General Relativity: Principles and Theorems You are in a small elevator. Is it possible to tell if you are accelerating downward without gravity or near a gravitating mass? 1. No. 2. Yes. 3. Only if the elevator eventually stops.

General Relativity: Principles and Theorems 1. No. In GR, pure linear acceleration is indistinguishable from gravity. This is a statement of the equivalence principle.

General Relativity: Principles and Theorems You are in a small elevator. Can you tell the difference between being near a gravitating mass or in a space station spinning to produce "artificial gravity"? 1. No. 2. Yes. 3. Only if "small elevator" is a euphemism for "physics hell".

General Relativity: Principles and Theorems 2. Yes. The artificial "gravity" created by rotation is inherently different than gravity, although it may feel the same at first. One way to tell the difference is to rapidly spin some object. The object will try to maintain its original spin axis. Therefore, if the object precesses, you might be on a spinning space station.

General Relativity: Equivalence Principle A hammer and feather both fall at the same speed. Youtube video of Astronaut showing this:

General Relativity: Equivalence Principle "It is only when there is numerical equality between the inertial and gravitational mass that the acceleration is independent of the nature of the body. " - Einstein • Weak Equivalence Principle: o All objects fall the same in a vacuum.

General Relativity: Equivalence Principle • Strong Equivalence Principle o ALL experiments appear the same when done § in a gravitational field, or § in a linearly accelerating laboratory. o G should be the same in the early universe.

GENERAL RELATIVITY: INTERVALS Static, symmetric spacetime: ds 2 = g 11 dr 2 + g 22 dθ 2 + g 33 dΦ 2 +g 44 dt 2 ds 2 = 0 is a "null path" where photons fly.

GENERAL RELATIVITY: SPACETIME INTERVALS Two events happen in spacetime. They are separated in space by Δr and in time by Δt. • • • Time-like interval o o One event can affect the other c Δt > Δr o o One event cannot affect the other c Δt < Δr o o Light from one could affect the other c Δt = Δr Space-like interval Light-like interval

GENERAL RELATIVITY: SPACETIME MOTION Two events happen in spacetime. They are separated in space by Δr and in time by Δt. • Closed timelike curve (Δr = 0; |Δt| < 0) A curve that comes back to the same spatial point at an earlier time (or later time) o backwards time travel allowed in GR o need other physics to disallow? o will cover time travel in other lectures o

General Relativity: Energy Conditions • Energy is conserved locally in GR. o Consequence of time symmetry o Consequence of Noether's theorem • Energy is NOT conserved globally in GR. o Example: cosmological redshift -- where does the energy go? o Potential energy is really just a bookkeeping device. § Not defined generally in GR.

General Relativity: The Key Equation Principle Equation: • Guv is the Einstein tensor o shows how spacetime curves • guv is the metric tensor o shows spacetime geometery • Tuv is the Stress Energy tensor o shows energy and energy flow • G, c, Λ are all constants

General Relativity: Curved Space • Flat Space o familiar Euclidean geometry holds o circles have area = π r 2 o spheres have volume = (4/3) π r 3 • Space curved by energy in GR o Euclidean geometry does NOT hold o Open space: § Area > π r 2; Volume > (4/3) π r 3 o Closed space: § Area < π r 2; Volume < (4/3) π r 3

General Relativity: Curved Space Example: Triangles Top: Triangle area is less than Euclidean Middle: Triangle area is more than Euclidean Bottom: "Flat. " Triangle area is exactly Euclidean.