string theory general relativity a new look at gravitation

String Theory

General relativity

A new look

at gravitation.

String Theory

• Recall the main points of the special relativity:

– Relativity of Simultaneity– Time Dilation– Length Contraction– Speed of light as a maximum speed.

String Theory

It’s the Law!

• Absolutely nothing can move faster than light c = 300,000 km/sec

• All observers agree on this speed, regardless of their own motion.

time

space

String Theory

Time dilation

Stationary observer at position A sends flash of light to observer and mirror at

position B, moving at velocity v. Time to send and receive is 2a/c.

Moving observer at B sees A moving to the left between A’s sending and receiving the (reflected) light, and so finds a longer path length of travel. If the speed of light is c,

the time also has to be longer.

String Theory

Relativity of Simultaneity

• Two observers in relative motion can disagree whether the event A is earlier than the event B

•

space

Light cone

Particle world line

time A B

String Theory

….Enter: Quantum Physics

String Theory

• Quantum Physics= Physics of quanta, the sub-atomic particles.

• Before Einstein, it was believed that the laws of kinematic were the same for all object, no matter how big or small are

String Theory

• … But think about the practical problems of measuring, for example, the position or the velocity of a very small particle…

String Theory

Heisenberg Uncertainty Principle

“The more precisely the position is determined, the less precisely the velocity is known in this instant, and vice versa”.

Werner Heisenberg (1901-1976) was a German theoretical physicist who made foundational contributions to quantum mechanics.

String Theory

• This is because any attempt to measure either the position or the velocity of a very small particle affect the

particle in question.

String Theory

How the speedometer cable attaches to the wheel of a bike

• The device that measures the speed of your car or motor bike interferes, even if slightly, with the motion of the wheel … so the devices that measure the motion of tiny particles will have to have some effect on them.

String Theory

• Imagine that you're blind and over time you've developed a technique for determining how far away an object is by throwing a medicine ball at it. If you throw your medicine ball at a nearby chair, the ball will return quickly, and you'll know that it's close. If you throw the ball at something across the street from you, it'll take longer to return, and you'll know that the object is far away.

String Theory

• The problem is that when you throw a ball -- especially a heavy one like a medicine ball -- at something like a chair, the ball will knock the chair across the room, and may even have enough momentum to bounce back. You can say where the chair was, but not where it is now. What's more, you could calculate the velocity of the chair after you hit it with the ball, but you have no idea what its velocity was before you hit it. 

String Theory

• This is the problem revealed by Heisenberg's Uncertainty Principle. To know the velocity of a particle we must measure it, and to measure it, we are forced to affect it. The same goes for observing an object's position. Uncertainty about an object's position and velocity makes it difficult for a physicist to determine much about the object.

• Of course, physicists aren't exactly throwing medicine balls at quanta to measure them, but even the slightest interference can cause the incredibly small particles to behave differently.

•

String Theory

RELATIVITY AND COSMOLOGY

Special relativity deals with unaccelerated motions (moving in a straight line at constant velocity) whereas general relativity deals with accelerated motions.

The more comprehensive General Theory of Relativity also includes a theory of gravitation.

Relativity requires four dimensions (three space dimensions, and time) to display (called “spacetime”), and so is somewhat difficult to conceptualize pictorially.

String Theory

A new look at mass and gravity

• Galileo and Newton stated that the “inertial mass” and “gravitational mass” of any object are equal. To see what this means, think about Newton’s Second Law. This relates the force exerted on an object to the acceleration it undergoes, setting them proportional to each other with the constant of proportionality being the inertial mass mi:

String Theory

• The inertial mass measures the resistance you feel when you try to push on the object; it is the same constant no matter what kind of force is being exerted.

• We also have the law of gravitation, which states that the gravitational force exerted on an object is proportional to the gravitational acceleration. The constant of proportionality in this case is called the gravitational mass mg:

• Fg = −mg .∇

String Theory

mg apparently has a very different character than mi; it is the “gravitational charge” of the body, and measures the effect of the gravitation on it.

Galileo showed (apocryphally by dropping weights off of the Leaning Tower of Pisa,) that the response of matter to gravitation was universal — every object falls at the same rate in a gravitational field, independent of the composition of the object. That is,

mi = mg for any object. An immediate

consequence is that the behavior of freely-falling test particles

is universal, independent of their mass

String Theory

• Imagine that we consider a physicist in a tightly sealed box, unable to observe the outside world, who is doing experiments involving the motion of test particles, for example to measure the local gravitational field. Of course she would obtain different answers if the box were sitting on the moon or on Jupiter than she would on the Earth.

String Theory

• But the answers would also be different if the box were accelerating at a constant velocity; this would change the acceleration of the freely-falling particles with respect to the box.

Accelerating Elevator

String Theory

String Theory

• Film clip from Nova/PBS• http://www.pbs.org/wgbh/nova/einstein/rela-i.html

String Theory

• By Galileo and Newton, there is no way to disentangle the effects of a gravitational field from those of being in a uniformly accelerating frame, simply by observing the behavior of freely-falling particles. This follows from the universality of gravitation.

•

String Theory

To be careful, we should limit our claims about the impossibility of distinguishing gravity from uniform acceleration by restricting our attention to “small enough regions of space-time.”

• If the sealed box were sufficiently big, the gravitational field would change from place to place in an observable way, while the effect of acceleration is always in the same direction. In a rocket ship or elevator, the particles always fall straight down:

String Theory

E= mc^2• After the advent of special

relativity, it became clear that mass was simply a manifestation of energy and momentum. It was therefore natural for Einstein to think about that there should be no way whatsoever for the physicist in the box to distinguish between uniform acceleration and an external gravitational field, no matter what experiments she did (not only by dropping test particles).

• This reasonable extrapolation became what is now known as the Einstein Equivalence Principle

• “In small enough regions of space-time, the laws of physics reduce to those of special relativity; it is impossible to detect the existence of a gravitational field.”

String Theory

But what if we are in a large region of space-time?

String Theory

Recall the Lorentz transformations…

• The Lorentz transformation for inertial frames can be shown to be:

–

• where    is called the Lorentz factor.

String Theory

• A visualization of the Lorentz transformation (full animation). Only one space coordinate is considered. The thin solid lines crossing at right angles depict the time and distance coordinates of an observer at rest with respect to that frame; the skewed solid straight lines depict the coordinate grid of an observer moving with respect to that same frame.

String Theory

We can imagine that the sections of a 4 dimensional space time are cones

• An observer can only detect “events” that occur within his or her “light cone”, whose boundaries correspond to velocities equal to the speed of light.

• Time can change only in one direction (forward, in the light cone diagram).

String Theory

The “natural” distance between two points

(t, x,y,z) in the cone space-time is

√(-Δt² +Δx²+Δy²+Δz²)

The model of the space-time is not the Euclidean model, but rather the Lobacevsky model of the hyperbolic geometry!

String Theory

… except that instead of a 3D space we have a 4D space …

String Theory

Shortest path in curved spacetime

String Theory

Remember the triangles in this model ?

•

String Theory

So, the space-time is then curved… and according to Einstein, the presence of planets make the curvature more

pronounced.

String Theory

• Like when you are sitting on a exercise ball …

String Theory

• As a consequence, in General Relativity, gravity is not a force, but it is attributed to the curvature of space and time.

String Theory

Newton couldn’t explain what gravity was. He thought of it as instantaneous action at a distance. Instead it can be viewed as an effect of the curved space-time.

• Mass tells space-time how to

curve

• Space tells mass how to move

• This naturally explains the

Universality of Free Fall

Acceleration – All objects move

along the same geometrical

distortions

•

String Theory

String Theory

• The curvature of the space-time is real! It has been measured with an orbiting gyroscope (Gravity probe B, and others)

• The gyroscope tips because the space is curved, just like the arrow before

Lgyroscope

axis rotates

String Theory

String Theory

And if we accept the principle that the light always

follows the fastest path between two points …

String Theory

… Then we have to accept that the gravity bends the rays of light

String Theory

This has been confirmed by the observation of the

precession of Mercury’s orbit

String Theory

And by the1919 eclipse…

String Theory

• The solar eclipse made possible to measure the displacement of one of the closest stars during the duration of the eclipses. The displacement was not as expected, but the discrepancy could be explained by Einstein’s general relativity theory: because the rays of light emanating from the star were not straight, but curved.