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Space Time Theory

Why no gravitational force in 3 dimensions?

Dipole radiation Quadrupole radiationElectromagnetic radiation is well-described by a wave equation where the basic mode of radiation is called the dipole, as pictured in the figure to the left. A dipole wave only needs one space direction to oscillate in, plus one direction to travel in, in addition to time. Therefore electromagnetic radiation can be described mathematically in 2+1 spacetime dimensions.
If we solve the Einstein equation and look for solutions that give gravitational radiation solutions, we find that the lowest wave mode of oscillation for gravitation radiation is the quadrupole, as pictured in the figure to the right.
But a quadrupole wave needs two space directions to oscillate in, and if we only have two space dimensions in our spacetime, then the wave still needs one more direction to travel in. So the lowest dimension spacetime where gravitational radiation is possible according to General Relativity is 3+1 dimensions. And that happens to be the number of spacetime dimensions we measure in our world.
The implications of this are that curvature in 2+1 spacetime dimensions can only exist locally in regions where matter is present. I.e that means that for the example of a single point mass, the spacetime everywhere around the mass will be flat according to the Einstein equations.
But curvature can still be measured in a spacetime with 2+1 dimensions with a point mass. An observer will feel no gravitational force from the mass itself, because force has to be transmitted causally and that means by a wave equation, and we don't have that here. But an observer who travels a closed path around the point mass can measure the total curvature located at the point mass, and we'll show that in detail in the next section.


Flat d = 2+1 spacetime 

Space paths

Light travels in straight lines without gravity. Here is a plot of the paths in space of three light rays passing through the space point x=1, y=0. In this plot, time is a parameter along the path, increasing radially outward.

Light rays

Geometry of d=3 with one point mass, according to Einstein equations

The solution to Einstein's equations for a three-dimensional spacetime with a single point mass looks like a flat spacetime everywhere to local observers -- except when they make measurements around spatially closed curves (curves that come back to the same place, but at a later time) that contain the mass itself.
When they do this, they learn that the space part of the spacetime they live in is not a flat plane, but a cone.

  Low mass

A Flatlander travelling around the blue arc will think she's been around a complete circle. But she will measure the circumference of this circle to be smaller than 2 R Pi, or 6 Pi in this case. The angular size of the missing wedge above is Pi/2, so the Flatlander will measure the circumference of her path to be 3 R Pi/2, or 4.5 Pi. Therefore she will be able to deduce that her path has encircled a point mass, and that she must live on a cone, not on a flat plane. Note that circular paths that do not circle the mass will have the normal circumference of 2 R Pi
The missing angle is called the deficit angle. According to Einstein's equations, the mass M of the point is related to the deficit angle B through the formula B=8GM Pi, where G is Newton's constant. The deficit angle can't be larger than 2Pi, so there is a limit on the allowed mass: M must be less than or equal to 1/4G. (Note: the units of Newton's constant are not the same in three and four spacetime dimensions. Why not?)
Another way of picturing this cone is shown below. It's just the xy plane with a wedge removed, and the sides sewn back together. Note that the blue circular arc is actually a closed path when the red dashed lines are identified by the sewing-together operation.




Space paths in flat coordinates

As we've gone over before in previous sections, in one time and two space dimensions, if we use the Einstein equations to obtain the spacetime metric for the spacetime around one massive point particle, all the curvature is concentrated where the mass is and the spacetime around the point mass is flat but with a deficit angle, as depicted in the figure below.
We see light flashes 1 and 2 leave from a common point at the same time until they cross where we've put the deficit angle gap. This crossing looks strange in flat coordinates, it looks like the flashes bounce from the gap and change identities.
Path for low mass


Space paths in smooth coordinates

If we get rid of the deficit angle by transforming to another set of coordinates where the angle is rescaled by that amount, we get a smooth picture that doesn't have to be sewn together across some angular gap, but the null geodesics in these new coordinates no longer look like straight lines, as shown below:

  Low mass


Spacetime paths in smooth coordinates

If we look at the paths we've seen in the last two sections in spacetime, where time has its own coordinate axis, the picture for low mass expands to what we see below.
Notice that the places where null geodesics cross are marked as points conjugate to the event t=0, x=1, y=0 through which they all pass initially. This behavior of null geodesics reconverging or crossing after emanating from a common source is of crucial importance to all the odd and interesting behaviors that can happen in the Genreal Relativity model of Nature. Black holes, naked singularities, clothed singularities, wormholes, time travel -- they all involve either an encouragement or a frustration of this null-geodesic-crossing phenomenon.


Y view


Light cones in smooth coordinates

Here is the light cone of our event in question, which is the surface made by all of the paths of light or null geodesics as we call them, that pass through the spacetime event t=0, x=1, y=0.
Now let's compare this with the previous of the light cone in flat spacetime. (Bottom Picture.)
Notice that the light cone below has big extra flap hanging on the inside. This extra flap is made of all the null geodesics that crossed a second time after leaving the event t=0, x=1, y=0, and then veered away from the main body of the light cone after that second crossing. Running down the middle of the extra flap, where the flap meets the normal part of the light cone, is the line of points conjugate to the original crossing event t=0, x=1, y=0.




light cone in flat spacetime.



What is the causal structure?

Causal structure of this spacetime

The deflection of light by gravity (such as it happens in General Relativity) does not blur at all the crisp and wide boundary between the past and future shown in the figure below. We can see quite well below what region of this spacetime is contained within the boundary of the causal future of the event E, what region of this spacetime is contained within the boundary of the causal past of the event E and what regions are outside of both the past and the future of this event. Causal structure shown by light cone