General Relativity is the title provided for Einstein’s concept of gravity that was explained in Chapter 2. As the theory is often presented, it describes gravity as a curvature in four-dimensional space-time. Today this is a principle way past the grasp of ordinary folks. Only the idea of four-dimensional space-time makes most of us to shudder … The answer in Quantum Field Theory is straightforward: Space is space and time is time, and there is no curvature. In QFT gravity is a quantum field in ordinary three-dimensional space, just like the other three force fields (EM, strong and weak).

This does not imply that four-dimensional notation is not helpful. It is a hassle-free way of handling the mathematical connection between space and time that is needed by special relativity. One may almost claim that physicists could not live without having it. Nonetheless, spatial and temporal evolution are fundamentally different, and I say shame on those that attempt to foist and force the four-dimensional idea onto everyone as vital to the understanding of relativity theory.

The idea of space-time curvature similarly had its inception in mathematics. When searching for a mathematical approach which could express his Principle of Equivalence, Einstein was brought to the equations of Riemannian geometry. And of course, these equations describe four-dimensional curvature, for individuals who can visualize it. You see, mathematicians are certainly not limited by physical restrictions; equations that possess a physical meaning in 3 dimensions can be generalized algebraically to any quantity of dimensions. But whenever you carry this out, you are certainly utilizing algebra (equations), not geometry (spatial configurations).

By stretching our minds, a few of us can even form a faint mental image of what four-dimensional curvature would be like in the event that it did exist. Nevertheless, saying that the gravitational field equations are equivalent to curvature is not the same as claiming that there is curvature. In QFT, the gravitational field is simply one more force field, like the EM, strong and weak fields, though with a more significant intricacy that is demonstrated in its higher spin value of 2.

While QFT resolves these paradoxical declarations, I do not want to leave you with the impression that the theory of quantum gravity is problem-free. While computational problems including the EM field were overcome by the procedure called renormalization, related problems involving the quantum gravitational field have not been overcome. Fortunately they do not conflict with macroscopic calculations, for which the QFT formulas become identical to Einstein’s.

Your choice. Once more you the reader have a choice, as you did in regard to the 2 approaches to special relativity. The choice is not actually about the equations, it is about their interpretation. Einstein’s equations may be deciphered as suggesting a curvature of space-time, unpicturable as it may be, or as describing a quantum field in three-dimensional space, much like the other quantum force fields. To the physicist, it truly doesn’t make much difference. Physicists are a lot more concerned with solving their equations rather than with deciphering them. If you would allow me one more Weinberg quote:

“The important thing is to be able to make predictions about images on the astronomers photographic plates, frequencies of spectral lines, and so on, and it simply doesn’t matter whether we ascribe these predictions to the physical effects of gravitational fields on the motion of planets and photons or to a curvature of space and time.” (The reader must be cautioned that these views are heterodox and would meet with arguments from several general relativists.)– Steven Weinberg

Therefore in case you want, you may believe that gravitational results are due to a curvature of space-time (even if you can’t picture it). Or, like Weinberg (and me), you can look at gravity as a force field which, like the various other force fields in QFT, occurs in three-dimensional space and develops over time according to the field equations.

More on Space-Time Curvature here