When you toss a coin, the coin falls down because there is more time at lower altitudes, and time passes slightly more slowly. If we plot altitude against time on a sheet then we expand time horizontally at lower heights, the sheet will splay outwards as shown below. A coin tossed upwards simply follows a locally straight path on this sheet, like an ant walking along it in a straight line.

The field equation of relativity holds at all points in space and time and is

where

*G*ij is the Einstein tensor describing the bending of space-time,

*k*is a constant and

*Tij*is the stress-energy tensor which represents the density and pressure of matter at that point.

*Gij*and

*Tij*are both 4x4 tensors (very similar to matrices) but they are symmetric and this means they represent a scaling along four orthogonal principle axes. For the spatial dimension this can be seen as stretching a sphere into an ellipsoid:

If we work in this principle inertial frame (the rest frame) then the field equation just has diagonal entries, e.g:

ρ is the density of matter and px,y,z are the pressures, which are equal in most idealised cases, such as an idealised fluid.

Scaling by

*k*gives the Einstein tensor

*Gij*, where each diagonal entry represents the scalar curvature of the 3D volume orthogonal to that entry's axis. For example the first entry

*means the 3D spatial curvature is*

*k*

*ρ*since the spatial volume is orthogonal to the time axis.

The 3D scalar curvature (or 3D Ricci scalar) is proportional to the difference between the surface area of a small sphere around that point, compared to the expected surface area of a sphere in Euclidean space. A positive scalar curvature (as is the case when mass is positive) has a smaller than expected surface area, and parallel paths converge. In the case of the first entry this means that mass causes space to have spherical curvature, like a 3-sphere.

This is the simplest explanation of General Relativity; the stress-energy tensor represents the scalar curvature of the 3D volume orthogonal to each principle axis.

### A closer look

However, this definition isn't very helpful in telling us how space-time actually distorts, since the scalar curvatures on each axis combine together in their effects. In order to get actual differential equations we expand out the field equation definition to:This transform can be visualised in 2D when the pressure is equal in all directions:

For any given mass and pressure of a body the blue arrows show the directions of increasing contraction with respect to time and with respect to space. Lets look at this curvature more carefully.

Each element in

*Rii*defines the rate of change of contraction of a small orthogonal 3D volume*V*with respect to the element's axis.
For

*R00*this 3D volume is just the spatial volume (written as V*xyz*):
\frac{\frac{d^2}{dt^2}V_{xyz}}{V_{xyz}}=-\frac{1}{2}(\rho+p_x+p_y+p_z)

as given by John Baez's relativity tutorial.

For

*R11*it is a small time period*t*multiplied by the perpendicular area*yz*:
\frac{\frac{d^2}{dx^2}V_{tyz}}{V_{tyz}}=-\frac{1}{2}(\rho+p_x-p_y-p_z)

and equivalently on the other two spatial axes. This is for a volume at rest, otherwise we subtract volume acceleration in flat space from the left hand side. The negative sign is because these are contractions.

So far we have used normalised units where the speed of light is 1, this is equivalent to using light-seconds (300,000 km, or distance to the moon) as the measure of distance. So for standard units the mapping looks more like:

In other words, effects are mostly due to mass density rather than pressure, and this produces mostly spatial volume contraction with respect to time, rather than significant changes over space.

### Example of earth:

Earth's mass is 6e24 kg and volume 1e21 m^3, so its density is: 6,000 kg/m^3

average pressure = 200e9 Pascals

### Example of space above earth:

The Ricci tensor in space is zero. This means no volume

- With respect to height, the time period of a volume expands (due to gravitational time dilation) but its width and length contract as horizontal area is slightly contracted at lower altitudes.

- With respect to time an initially static volume's height expands since gravity is slightly stronger closer to earth, but its width and length get shorter, as it moves downward where area is slightly contracted.

*V**xyz*acceleration with time, and no time x area (*Vtyz*) acceleration with distance x. This does not make space Lorentz flat, it can still bend and preserve 3-volume and that is what it does near earth:- With respect to height, the time period of a volume expands (due to gravitational time dilation) but its width and length contract as horizontal area is slightly contracted at lower altitudes.

- With respect to time an initially static volume's height expands since gravity is slightly stronger closer to earth, but its width and length get shorter, as it moves downward where area is slightly contracted.

In both cases the expansion and contractions cancel out to give zero volume change.

This diagram shows how object move above ground, causing vertical expansion, and below ground, causing vertical contraction. In solids the pressure forces resist the inwards pull. Therefore relative to these straight yellow lines, the ground surface is accelerating upwards.

Spatial curvature is much less pronounced than curvature with time. This bend is irrespective of speed.

Spatial curvature is much less pronounced than curvature with time. This bend is irrespective of speed.

TODO:

2. earth is 2mm proper radius (bend) wider than if space was Euclidean

3. sun is 2km wider than if space was Euclidean, which is 6ppm (six earths) larger volume

5. are grav waves due to zero divergance constraint (which pulls 10 dofs down to 6), or due to Levi Vita connection, or the Bianchi identities?

For more details try out my cheat sheet:

and some helpful links: