Geometry of space-time

Now we know, that

• in flat space, the shortest distance between two points is a straight line.
• in curved space, the shortest distance between two points is not necessarily a straight line.
• in flat (i.e. Minkowski) space-time inertial observers move between two events along straight lines in such a way as to maximize their proper time.

Thus, we can conclude, that

• in curved space-time inertial observers move between two events not necessarily along straight lines, but in such a way as to maximize their proper time.

This type of world line is called a geodesic.

Tidal forces

The scale of a given inertial frame is limited by tidal force s. Since the gravity forces changes with distance, a force acting on one part of an object may be different from the force acting on another part of an object.

The tidal forces are responsible for the tides on Earth, because the Moon gravity is different at two sides of the Earth. The water in the ocean bulges on both sides of the Earth.

It is clear why the water bulges on the side of the Earth facing the Moon: the Moon gravity pulls it. But why does the ocean bulge on the other side of the Earth?

A:

The Sun gravity pulls it the other way.

B:

The Moon gravity becomes antigravity beyond the Earth center.

C:

The Moon gravity is weaker on the farthest side of the Earth then at the Earth center.

Tidal forces were responsible for the break-up of Shoemaker-Levi comet when it fell into Jupiter.

The existence of the tidal forces means that in the gravitationa l field, different places within one inertial frame feel slightly different gravity forces, whereas in an accelerated frame all places move with precisely the same acceleration.

Thus, the equivalence principle is not exact - it is only approximate for any object of finite size. It becomes exact only for a point.

It is said that inertial frames in GR are local, i.e. the y only extend for a very small distance around an observer.

Einstein equations of gravity

After some 10 years of continuous search, Einstein finally arriv ed at the equation(s) of GR (Einstein's gravity law):

Here G_ij is called the Einstein's tensor, it describes the geometry of the space-time. T_ij is called the stress-energy tensor, it describes the matter and energy in this space-time.

Symbolically, we can write Einstein equations in the following f orm:

$$\mbox{\sl GEOMETRY} = \mbox{\sl MATTER}+\mbox{\sl ENERG Y}$$

In its complete mathematical form, they look like this:

$${1\over2}g^{rs}\left( - {\partial^2g_{ij}\over\partial x^r\pa... ...} - {\partial^2g_{rs}\over\partial x^i\partial x^j}\right) +$$

$${1\over4}g^{qp}\left( -{\partial g_{is}\over\partial x^p} + ... ...er\partial x^j} - {\partial g_{rj}\over\partial x^q}\right) -$$

$${1\over4}g^{qp}\left( - {\partial g_{ij}\over\partial x^p} + ... ...er\partial x^r} - {\partial g_{rs}\over\partial x^q}\right) -$$

$${1\over4}g_{ij}g^{rs}g^{uv}\left( - {\partial^2g_{rs}\over\pa... ...} + {\partial^2g_{us}\over\partial x^r\partial x^v} - \right.$$

$$\left. {\partial^2g_{uv}\over\partial x^r\partial x^s}\right)... ...artial x^r} - {\partial g_{rv}\over\partial x^q}\right)\times$$

$$\left( {\partial g_{ps}\over\partial x^u} + {\partial g_{pu}... ...^{uv}g^{qp}\left( {\partial g_{qr}\over\partial x^s} + \right.$$

$$\left. {\partial g_{qs}\over\partial x^r} - {\partial g_{rs}... ...ial g_{uv}\over\partial x^p}\right) = {8\pi G\over c^4}T_{ij}.$$