Thursday, July 13, 2017

general relativity

When you toss a coin, the coin falls down because there is less 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. 
We stretched rather than contracted the time direction because we are representing the time dimension spatially, and they have opposite metric signatures. Space and time don't actually bend into a third dimension like the sheet, the curvature is just intrinsic to the space and time; you can just think of time as being less dense at lower altitudes.

The field equation of relativity holds at all points in space and time and is
where Gij 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 kρ since the spatial volume is orthogonal to the time axis.

This 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 sphere 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.

For earth, the density ρ is 5,500 kg/m^3 and since k is the rather small 2e-43, the spatial curvature G00 is 1.1e-39. This results in the earth being 2mm smaller in radius and with 35 fewer hectares of ground than if it were in flat space. 

A closer look

However, this definition isn't very helpful in telling us how space and time distort with each other, 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:

where Rij is the Ricci tensor which represents the 4D curvature of space-time and R is the 4D Ricci scalar. Working in the mass's rest frame, the Ricci tensor is just a linear transform of the stress-energy tensor:

This transform can be graphed in 2D when the pressure is equal in all directions (using units where k=1):

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 Vxyz): 
as given by John Baez's relativity tutorial. For R11 it is a small time period t multiplied by the perpendicular area yz:
Note that Vtyz isn't exactly a volume, while increases in the yz directions increase it, increases in the time period do the opposite, due to the negative metric signature:
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.

In standard units the equations just contain a few more constants:
Noting that the first term in the stress-energy tensor is actually energy density rather than just density, and since e=mc^2, that is why we use ρc^2.
So for these "normal" units of density and pressure, the pressure has almost zero effect and also changes with respect to space are tiny compared to those with respect to time. 
For example, for earth ρ is 5,500 kg/m^3 and earth's pressure is 2e11 Pascals, so the pressure term is 4e-8% of the energy density term. 
So the rate of volume contraction is 0.0005% per second squared. 
whereas the rate of Vtyz contraction with distance is 5e-21% per metre squared.

Therefore curvature per metre squared is roughly 1e-15% of the curvature per second squared, for earth.

Example of space above earth:

The Ricci tensor in space is zero. This means no volume Vxyz  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 horizontal area contracts at lower altitudes (being 35 hectares less than in flat space at ground level), and the other coordinate must expand to compensate (we draw it spatially as expanding), but since this is the negative signature time period, it must also reduce, and it does, causing gravitational time dilation.
- 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 objects 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 in fact accelerating upwards.

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

For more details try out my cheat sheet:

and some other helpful links: