Introduction to General Relativity Part I Outline Brief

Introduction to General Relativity Part I

Outline • Brief Introduction • Basics of General Relativity • Gravity Probe B

Introduction Newton’s Universal Law of Gravitation • 2 masses m 1 & m 2 attract each other with a force F that is proportional to the product of their masses & inversely proportional to the square of the distance r between them.

Special Relativity • As we’ve been discussing, Einstein published his first paper on special relativity in 1905. • He started with 2 postulates: – Postulate 1: The laws of physics are the same for all inertial observers – Postulate 2: The speed of light is the same for all inertial observers

Conflict! • No signal can travel faster than light, but in Newton's theory, Gravity is a Force that is transmitted instantaneously over vast distances. • Example: Masses at the ends of the universe • So, the question is: What is wrong with Newton’s gravitational force?

Basics of General Relativity Mach’s Principle • Newtonian inertial frames can not be rotating with respect to the “fixed” stars. • Inertial properties of a reference frame are determined by the presence of other bodies in the universe.

Principle of Equivalence “We shall therefore assume that there is complete physical equivalence of a gravitational field the corresponding acceleration of the reference frame. ” • This assumption extends the principle of relativity to the case of the uniformly accelerated motion of the reference frame!

• General relativity is an extension of special relativity. It includes the effects of accelerating objects & their masses on spacetime. • As a result, theory is an explanation of gravity. It is based on two concepts: 1. The Principle of Equivalence of the heavy mass & the dynamic mass, which is “kind of” an extension of Einstein’s first postulate of special relativity & 2. The curvature of spacetime due to gravity.

The Principle of Equivalence: • There is no physical experiment that can detect the difference between a uniform gravitational field an equivalent uniform acceleration.

The Principle of Equivalence • The Principle of Equivalence can be shown by experiments in an inertial reference frame. • Consider an astronaut sitting in a confined space on a rocket placed on Earth. • The astronaut is strapped into a chair that is mounted on a weighing scale that indicates a mass M. The astronaut drops a safety manual that falls to the floor.

The Principle of Equivalence • Now, contrast this situation with the rocket accelerating through space. The gravitational force of the Earth is now negligible. • If the acceleration due to gravity has exactly the same magnitude g as on Earth, then the scale will read the same mass M that it did on Earth, & the safety manual will fall with the same acceleration as measured by the astronaut. The question is: How can the astronaut tell whether the rocket is on earth or in space?

The Principle of Equivalence • Now, contrast this situation with the rocket accelerating through space. The gravitational force of the Earth is now negligible. • If the acceleration due to gravity has exactly the same magnitude g as on Earth, then the scale will read the same mass M that it did on Earth, & the safety manual will fall with the same acceleration as measured by the astronaut. The question is: How can the astronaut tell whether the rocket is on earth or in space? Answer: There is no way to tell!

Inertial Mass & Gravitational Mass • Recall from Newton’s 2 nd Law that an object accelerates in reaction to a force according to its Inertial Mass m. I. • Inertial Mass m. I is a measure of how strongly an object resists a change in its motion: as described by Newton’s 2 nd Law:

Inertial Mass & Gravitational Mass • Gravitational Mass m. G is a measure of how strongly an object attracts another object through the gravitational force of attraction between them, as defined by Newton’s Universal Law of Gravitation:

Inertial Mass & Gravitational Mass • For the same force, we obviously can get a ratio of the Gravitational Mass m. G to the Inertial Mass m. I: • According to the Principle the Inertial of Equivalence, Mass m. I & the Gravitational Mass m. G are equal. So, a = g

Principle of Covariance • In Special Relativity, all inertial observers equivalent. • In General Relativity, all observers observe the same laws of physics. The Laws of Physics can be expressed in terms of tensors, since tensors are defined independent of any coordinate system.

Correspondence Principle • For masses in weak gravitational fields moving with velocities v << c • The General Theory should agree with Newtonian Mechanics • As gravitational fields go to zero, the The General Theory should agree with Newtonian Mechanics

The Principle of Minimal Gravitational Coupling • No terms explicitly containing the curvature of space time should be added when going from Special Relativity to General Relativity

• This leads to the idea that space-time is curved The Field Equation is shown below. Rik Ricci Curvature Tensor R Scalar Curvature gik Metric Tensor Λ Cosmological Constant Tik Stress-Energy Tensor describing the non-gravitational matter, energy & forces at any given point in space-time

Basics of General Relativity • Space & time are warped by the presence of the Earth, & the Earth's rotation drags space-time around with it!

Gravity Probe B Proposed by Stanford U Physicist Leonard Schiff • The experiment measures how space & time are warped by the presence of the Earth, & how the Earth's rotation drags space-time around with it.

Gravity Probe B • A gyroscope (actually four) and telescope will be placed in a satellite and sent into a polar orbit 400 miles (640 km) above the Earth. • The telescope & the gyroscope’s spin axis will be perfectly aligned, and both will point at a distant star (IM Pegasi).

Gravity Probe B • Classically, the underlying principle of a gyroscope is that rotating systems, free from disturbing forces, should have their anglar momentum stay pointing in the same direction in space. However, as mentioned before, space-time is warped & may even be set in motion by moving matter. • A gyroscope orbiting the Earth will find two space-time processes acting on it frame-dragging & the geodetic effect gradually changing its direction of spin.

Gravity Probe B Frame Dragging • In 1918, Josef Lense and Hans Thirring predicted that masses in space-time would not only curve the structure of space-time around them, but would also twist the local space-time around them if they are rotating • This twisting would alter measurements of distance, time and direction in local space-time. Example: Racquetball in honey!

Gravity Probe B The Geodetic Effect • When a mass “sits” in space-time, it causes the space-time “fabric” to deform. • This changes the space & alters the passage of time around that object. In the case of the Sun’s “dip”, as the planets travel across space-time, they respond to the “dip” and follow the curve in space-time and travel around the Sun.

This figure shows the Earth distorting Space-Time around it.

Gravity Probe B Note: Why IM Pegasi? • Stars are not fixed in the sky. • IM Pegasi was chosen because it is one of the brightest stars in the sky, in the microwave range. • Very-Long-Baseline Interferometry (VLBI) measurements since 1997 were used to map IM Pegasi’s motion in the sky.

Gravity Probe B Note: “Perfect” Gyroscopes • In order for GP-B to measure anything, its gyroscopes must be nearly perfect • Must not wobble or drift more than 1 e-11 degrees in an hour while its spinning, since the predicted twist in space-time is on the order of ~ -6 10 degrees each year.

“Perfect” Gyroscopes • Must be near perfect spherically & homogeneity • Gyro Size: 1. 5 inches of fused quartz • Sphericity: <40 atomic layers from perfect

“Perfect” Gyroscopes • Quartz Purity: Within 2 parts per million • Gyro Spin Rate: ~10, 000 rpm • Gyro Drift Rate: < 10 -12 degrees/hour

Gravity Probe B • Note: How do you measure the twist? Superconductivity • Gyroscopes coated with a sliver thin layer (1270 nm) of niobium. • As the gyroscope spins, the niobium layer generates a magnetic field, whose axis is exactly aligned with the spin axis.

Superconductivity • A thin superconducting metal loop encircles each gyroscope. • The loop is connected to an external SQUID (Superconducting Quantum Interference Device) • If and when the gyro tilts, the magnetic field changes with it, which affects the current in the loop

Superconductivity

Gravity Probe B • According to Einstein’s theory, the gyroscope should turn in two directions simultaneously. • As it travels through curved space time, it will turn slightly along one axis. • As “frame-dragging” occurs, it will also turn slightly along a perpendicular axis. • Schiff calculated that at a 400 -mile altitude, space time curvature would turn the gyroscope 6. 6 arcseconds per year in one direction, and the “frame-dragging ” will turn at. 042 arcseconds per year in a perpendicular direction.

Gravity Probe B

• To recap, conflicts between Newton’s Universal Law of Gravitation & Einstein’s Special Theory of Relativity lead to the formalization of General Relativity. • GP-B is in orbit, and is scheduled to orbit for a while longer