General Relativity is the title given to Einstein’s theory of gravitational force that was illustrated in Chapter 2 of my book. As the theory is normally shown, it illustrates gravity as a curvature in four-dimensional space-time. Now this is an idea far over and above the reach of regular individuals. Simply the idea of four-dimensional space-time causes most of us to tremble … 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 regular three-dimensional space, the same as the other 3 force fields (EM, strong and weak).
This does not indicate that four-dimensional notation is not useful. It is a practical approach of addressing the mathematical relationship between space and time which is needed by special relativity. One may almost say that physicists could not live without it. Nevertheless, spatial and temporal evolution are fundamentally different, and I say shame on those who try to pass off and force the four-dimensional idea onto the general public as important to the awareness of relativity theory.
The concept of space-time curvature likewise had its origin in mathematics. When looking for a mathematical method that could embody his Principle of Equivalence, Einstein was led to the equations of Riemannian geometry. And yes, these formulas explain four-dimensional curvature, for individuals who can easily visualize it. You see, mathematicians are certainly not restricted by physical restrictions; equations that have a physical meaning in 3 dimensions may be generalized algebraically to any variety of dimensions. But when you do this, you are definitely managing algebra (equations), not geometry (spatial configurations).
By stretching our minds, some of us are able to even create a faint mental image of what four-dimensional curvature would resemble if it did exist. Nonetheless, stating that the gravitational field equations are equivalent to curvature is certainly not the same as saying that there is curvature. In Quantum Field Theory, the gravitational field is just an additional force field, like the EM, strong and weak fields, albeit with an increased complexity which is shown in its higher spin value of 2.
While QFT resolves these paradoxical declarations, I really don’t wish to leave you having the thought that the theory of quantum gravity is problem-free. Whilst computational troubles concerning the EM field were overcome with process called renormalization, very similar challenges involving the quantum gravitational field have not been overcome. Thankfully they do not actually interfere with macroscopic calculations, for which the QFT formulas become identical to Einstein’s.
Your choice. Once again you the reader have a choice, as you did in concern to the two approaches to special relativity. The choice is not regarding the formulas, it is about their interpretation. Einstein’s equations can be translated as suggesting a curvature of space-time, unpicturable as it may be, or as explaining a quantum field in three-dimensional space, just like the other quantum force fields. To the physicist, it really doesn’t make much difference. Physicists are much more concerned with solving their formulas rather than with interpreting them. If you will permit me another 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 should be warned that these views are heterodox and would meet with objections from many general relativists.)– Steven Weinberg
Thus in case you prefer, you can think that gravitational effects are due to a curvature of space-time (even if you can’t picture it). Or, like Weinberg (and myself), you may see gravity as a force field that, like the other force fields in Quantum Field Theory, exists in three-dimensional space and progresses in time according to the field equations.
Learn more about space-time curvature at Fields of Color!