Special Topic: Relativity and Spacetime

Postulates of special relativity

1. The laws of nature are the same for everyone. 2. The speed of light is the same for everyone. • All of relativity follows from these two ideas!

Length contraction

l' = l*sqrt(1-(v^2/c^2))

Mass increase

m' = m / sqrt(1-(v^2/c^2))

Time dilation

t' = t * sqrt(1-(v^2/c^2))

Perspectives in Space

• A book has a definite three-dimensional shape. • But the book looks different in two-dimensional pictures of the book taken from different perspectives. • Similarly, space and time look different from different perspectives in spacetime.

Mass Increase

• A force applied to a rapidly moving object produces less acceleration than if the object were motionless. • This effect can be attributed to a mass increase in the moving object.

Rules of Spherical Geometry

• A great circle is the shortest distance between two points. • Parallel lines eventually converge. • Angles of a triangle add up to > 180°. • Circumference of circle is < 2pi*r.

Rules of Saddle-Shaped Geometry

• A piece of a hyperbola is the shortest distance between two points. • Parallel lines diverge. • Angles of a triangle add up to < 180°. • Circumference of circle is > 2pi*r.

Rules of Geometry in Flat Space

• A straight line is shortest distance between two points. • Parallel lines stay the same distance apart. • Angles of a triangle add up to 180°. • Circumference of a circle is 2pi*r.

Worldlines

• A worldline shows an object's path through spacetime in a spacetime diagram. - Vertical worldline: no motion - Diagonal worldline: constant-velocity motion - Curved wordline: accelerating motion

"Straight" Lines in Spacetime

• According to equivalence principle: - If you are floating freely, then your worldline is following the straightest possible path through spacetime. - If you feel weight, then you are not on the straightest possible path

Dimensions of Space

• An object's number of dimensions is the number of independent directions in which movement is possible within the object.

Making Sense of Relativity

• As children, we revised our ideas of "up" and "down" when we learned that Earth is round. • Relativity forces us to revise how we think of "space" and "time." • According to you, time slows down in a moving spaceship. • According to someone on that spaceship, your time slows down. • Who is right? • You both are, because time is not absolute but depends on your perspective.

Curvature Near Black Hole

• Continued shrinkage of Sun would eventually make curvature so great that it would be like a bottomless pit in spacetime: a black hole. • Spacetime is so curved near a black hole that nothing can escape. • The "point of no return" is called the event horizon. • Event horizon is a three-dimensional surface.

Thought Experiments

• Einstein explored the consequences of the absoluteness of light speed using "thought experiments." • The consequences will be easiest for us to visualize with thought experiments involving spaceships in freely floating reference frames (no gravity or acceleration).

The Equivalence Principle

• Einstein preserved the idea that all motion is relative by pointing out that the effects of acceleration are exactly equivalent to those of gravity.

Tests of Relativity

• First evidence for absoluteness of speed of light came from the Michelson-Morley experimentperformed in 1887. • Time dilation happens routinely to subatomic particles that approach the speed of light in accelerators. • Time dilation has also been verified through precision measurements in airplanes moving at much slower speeds. • Prediction that E= mc^2 is verified daily in nuclear reactors and in the core of the Sun.

Key Ideas of General Relativity

• Gravity arises from distortions of spacetime. • Time runs slowly in gravitational fields. • Black holes can exist in spacetime. • The universe may have no boundaries and no center but may still have finite volume. • Rapid changes in the motion of large masses can cause gravitational waves.

Curved Spacetime

• Gravity can cause two space probes moving around Earth to meet. • General relativity says this happens because spacetime is curved.

Acceleration and Relative Motion

• How can your motion be relative if you're feeling a force causing acceleration?

Special Topic: The Twin Paradox

• If one twin takes a high-speed round trip to a distant star, that twin will have aged less than the other that remains on Earth. • But doesn't time on Earth appear to run slower from the perspective of the twin on the high-speed trip? • Solution: The twin on the trip is accelerating. • The shortest path may look curved from some perspectives, but more time always passes for the twin following the shorter path through spacetime

Simultaneous Events?

• In your reference frame, red and green lights on other spaceship appear to flash simultaneously. • But someone on the other spaceship sees the green light flash first—simultaneity is relative!

How does gravity affect time?

• Light pulses travel more quickly from front to back of an accelerating spaceship than in other direction. • Everyone on the ship agrees that time runs faster in front than in back. • The effects of gravity are exactly equivalent to those of acceleration. • Time must run more quickly at higher altitudes in a gravitational field than at lower altitudes.

What would it be like to visit a black hole?

• Light waves take extra time to climb out of a deep hole in spacetime, leading to a gravitational redshift. • Time passes more slowly near the event horizon. • Tidal forces near the event horizon of a 3MSunblack hole would be lethal to humans. • Tidal forces would be gentler near a supermassive black hole because its radius is much bigger.

Limitations of the Rubber Sheet Analogy

• Masses do not rest "upon" the spacetime like they rest on a rubber sheet. • The rubber sheet shows only two dimensions of space. • Rubber sheet shows only two dimensions of space. • Path of an orbiting object actually spirals through spacetime as it moves forward in time.

What is relative about relativity?

• Motion is not absolute—we must measure speed of one object relative to another. • Example: A plane moving at 1670 km/hr from east to west would appear from space to be standing still.

Gravity, Newton, and Einstein

• Newton viewed gravity as a mysterious "action at a distance." • Einstein removed the mystery by showing that what we perceive as gravity arises from curvature of spacetime.

Key Ideas of Special Relativity

• No material object can travel faster than light. • If you observe something moving near light speed: - Its time slows down. - Its length contracts in direction of motion. - Its mass increases. • Whether or not two events are simultaneous depends on your frame of reference.

No Escape

• Nothing can escape from within the event horizon because nothing can go faster than light. • No escape means there is no more contact with something that falls in. It increases the hole mass, changes the spin or charge, but otherwise loses its identity.

Perspectives in Spacetime

• Observers in relative motion do not share the same definitions of x, y, z,and t,taken individually: - Space is different for different observers. - Time is different for different observers. - Spacetime is the same for everyone.

Rubber Sheet Analogy

• On a flat rubber sheet: - Free-falling objects move in straight lines. - Circles all have circumference 2pi*r. • Matter distorts spacetime in a manner analogous to how heavy weights distort a rubber sheet. • Mass of Sun curves spacetime: - Free-falling objects near Sun follow curved paths. - Circles near Sun have circumference < 2pi*r.

Relativity and Acceleration

• Our thought experiments about special relativity involved spaceships moving at constant velocity. • Is all motion still relative when acceleration and gravity enter the picture?

Length Contraction

• Similar thought experiments tell us that an object's length becomes shorter in its direction of motion.

Path of Ball in a Moving Train

• Someone outside the train would see the ball travel a longer path in one up-down cycle. • The faster the train is moving, the longer that path would be

Gravity and Relative Motion

• Someone who feels a force may be hovering in a gravitational field. • Someone who feels weightless may be in free-fall.

Einstein's Theories of Relativity

• Special Theory of Relativity (1905) - Usual notions of space and time must be revised for speeds approaching light speed (c). - E= mc^2 • General Theory of Relativity (1915) - Expands the ideas of special theory to include a surprising new view of gravity

Curvature Near Sun

• Sun's mass curves spacetime near its surface. • If we could shrink the Sun without changing its mass, curvature of spacetime would become greater near its surface, as would strength of gravity.

Why can't we reach the speed of light?

• Suppose you tried to catch up to your own headlight beams. • You'd always see them moving away at speed c. • Anyone else would also see the light moving ahead of you.

"Surface" of a Black Hole

• The "surface" of a black hole is the radius at which the escape velocity equals the speed of light. • This spherical surface is known as the event horizon. • The radius of the event horizon is known as the Schwarzschild radius. • The event horizon of a 3MSunblack hole is also about as big as a small city. • The event horizon is larger for black holes of larger mass. • A black hole's mass strongly warps space and time in the vicinity of its event horizon.

A Journey to Vega

• The distance to Vega is about 25 light-years. • But if you could travel to Vega at 0.999c, the round trip would seem to take only 2 years! • At that speed, the distance to Vega contracts to only 1 light-year in your reference frame. • Going even faster would make the trip seem even shorter! • However, your twin on Earth would have aged 50 years while you aged only 2 years.

Geometry on a Curved Surface

• The straightest lines on a sphere are great circles sharing the same center as the sphere. • Great circles intersect, unlike parallel lines in flat space. • Straight lines are the shortest paths between two points in flat space. • Great circles are the shortest paths between two points on a sphere.

Geometry of the Universe

• The universe may be flat, spherical, or saddle-shaped depending on how much matter (and energy) it contains. - Flat and saddle-shaped universes are infinite in extent. - Spherical universe is finite in extent. - No center and no edge to the universe are necessary in any of these cases.

Path of Ball in a Stationary Train

• Thinking about the motion of a ball on a train will prepare us for the next thought experiment.

Curved Space

• Travelers going in opposite directions in straight lines will eventually meet. • Because they meet, the travelers know Earth's surface cannot be flat—it must be curved.

Dimensions of Spacetime

• We can move through three dimensions in space (x, y, z). • Our motion through time is in one direction (t). • Spacetime, the combination of space and time, has four dimensions (x, y, x, t). • Special relativity showed that space and time are not absolute. • Instead, they are inextricably linked in a four-dimensional combination called spacetime.

Time Dilation

• We can perform a thought experiment with a light beam replacing the ball. • The light beam, moving at c, travels a longer path in a moving object. • Time must be passing more slowly there.

Worldlines for Light

• Worldlines for light go at 45°angles in diagrams with light-seconds on one axis and seconds on the other.