General Relativity Equivalence of Mass and Energy 0 + - = 9 8 7 1 2 3 4 5 6 c Tests of General Relativity Upon developing the general theory of relativity, Einstein predicted that the elliptical orbits of the planets precess about the sun, starlight passing close to the sun is deflected, and gravitation causes time to slow down

Later, his predictions were successfully tested and confirmed

Precession of the Planetary Orbits

Using four-dimensional field equations, Einstein recalculated the orbits of the planets about the sun

Deflection of Starlight

Einstein predicted that starlight passing close to the sun would be deflected by an angle of 1.75 seconds of arc

Gravitational Red Shift

Einstein predicted that gravity causes clocks to run slow

light traveling "against gravity" is observed to have a slightly lower frequency

Main Ideas Momentum and Inertia in Relativity

Equivalence of Mass and Energy

The Correspondence Principle

General Relativity

Gravity, Space, and a New Geometry

Tests of General Relativity Momentum and Inertia in Relativity If we push an object that is free to move, it will accelerate. If we maintain a steady push, it will accelerate to higher and higher speeds. If we push with a greater and greater force, we expect the acceleration in turn to increase.

The speed limit of the universe is the speed of light.

We are not able to accelerate any object fast enough to match the speed of light.

Newton's second law in terms of momentum:

F = mv/ t

Remains valid in relativity theory

To Newton, infinite momentum would mean infinite speed, which is not so in relativity.

Einstein's new definition of momentum:

p = mv/ 1 - v^2/c^2

As on object approaches the speed of light, its momentum increases dramatically

Rest mass: represented by m is a true constant, a property of the object no matter what speed it has Einstein concluded that it takes energy to make mass and that energy is released when mass disappears

Rest mass is a kind of potential energy

Mass stores energy

E = mc^2

Mass and energy are equivalent - anything with mass also has energy

A remarkable insight of Einstein's special theory of relativity is his conclusion that mass is simply a form of energy

Rest energy: "energy of being" of all pieces of matter even at rest and even not interacting with anything else Gravity, Space, and a New Geometry Space-time has four dimensions - three space dimensions (such as length, width, and height) and one time dimension (past to future)

Einstein perceived a gravitational field as a geometrical warping of four-dimensional space-time

Geodesics: lines of shortest distance when confined to a curved surface

The presence of mass produces a curvature or warping of space-time: conversely, a curvature of space-time reveals the presence of mass Chapter 16: Relativity - Momentum, Mass, Energy, and Gravity The Correspondence Principle Conceptual Physics The Correspondence Principle states that new theory and old must overlap and agree in the region where the results of the old theory have been fully verified

According to the correspondence principle, if the equations of special relativity (or any other new theory) are to be valid, they must correspond to those of Newtonian mechanics - classical mechanics - when speeds much less than the speed of light are considered

General theory of relativity: a new theory of gravitation, in which gravity causes space to become curved and time to slow down Einstein concluded, in what is now called the principle of equivalence, that gravity and accelerated motion through space-time are related

The principle of equivalence states that local observations made in an accelerated frame of reference cannot be distinguished from observations made in a Newtonian gravitational field

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# Chapter 16: Relativity - Momentum, Mass, Energy, and Gravity

Conceptual Physics