Topology of Relativity: Implications of Niels Bohr's Arguments

[victorvmotti]

I have seen in the online Stanford Encyclopedia of Philosophy in the entry on Copenhagen Interpretation of Quantum Mechanics that Niels Bohr had argued that the theory of relativity is not a literal representation of the universe:

"Neither does the theory of relativity, Bohr argued, provide us with a literal representation, since the velocity of light is introduced with a factor of i in the definition of the fourth coordinate in a four-dimensional manifold."

By this he means that like the quantum mechanics, the theory of relativity involves $$i$$ in its equations. That is in the fourth dimension of spacetime. Therefore it is a symbolic not pictorial representation of the universe.

Now I am trying to make sense of the geometrical implications for the spacetime.

Consider a flat universe and dimensions in which $$c=1$$. The metric with the $$-+++$$signature could be read like:

$$i^2d\alpha^2+d\theta^2+d\phi^2+d\psi^2$$

We can have the global spacetime topology defined by

$$C_{\infty} \times C_{\infty} \times C_{\infty} \times C_{\infty}$$

Where $$C_{\infty}$$ is the Riemann sphere.

Now the above local metric of a flat universe can be interpreted as describing a complex 4-torus or just one curve consisting of the product of four circles on the global product of four Riemann spheres. With time dimension, $$i \alpha$$, in the current universe, charted to only imaginary and the other dimensions to the real.

So globally, in this topology, we have this closed timelike curve, which is the only curve.

Am I right about the conclusions?

So my question is that what are the implications, especially the in terms of the insights related to the global topology of the universe, if we note that the theory of relativity is not a pictorial representation as highlighted by Niels Bohr.

 

[Nugatory]

So my question is that what are the implications, especially the in terms of the insights related to the global topology of the universe, if we note that the theory of relativity is not a pictorial representation as highlighted by Niels Bohr.

You've found some of the many reasons to dislike the $ict$ approach to describing spacetime geometry. This approach was fashionable back in its day, and that's likely when Bohr made that comment. It has the advantage of making the Lorentz transform in Minkowski space look formally like a rotation in Euclidean space; but this is more than negated by being both misleading and not easily carried forward into general relativity.

Unlike QM, introducing complex numbers into relativity is neither necessary or helpful - so no matter what they used to do back then, it's better not to.

As for whether relativity, or any physical theory for that matter, can be properly called a "pictorial representation"... That's a completely sterile discussion and a pretty good way of getting a thread locked.