Why no gravitational force in 3 dimensions?Electromagnetic 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.
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