General Relativity is pure Mathematics

[harrylin]

General Relativity Spacetime is pure Mathematics. But we live in a real world. How do we get coupled to Spacetime? We are real. Spacetime is math. Math just describe reality and doesn't affect it. So Spacetime only describe reality as metaphors. But how come physicists believe Spacetime is real. How can this model work when the manifold is just pure math and we who are real and solid can't interact with a manifold that has no physical form. Please explain. Thanks in advance.

Of course a description of observations of reality cannot affect it - but why would we want our descriptions to affect reality?

General relativity is just like other theories of physics, a theory about physical measurements, as done with, among other things, time and length standards. Spacetime is a mathematical description of a standard measurement system that consists of clocks and rulers (or equivalent). This is explained by Einstein in the introduction of his 1916 paper:

- http://www.Alberteinstein.info/gallery/gtext3.html

Near the end of it he also predicted consequences for the "real world".

He also partially answered your question in a popular account:

"rays of light are propagated curvilinearly in gravitational fields."

"the general theory of relativity enables us to derive theoretically the influence of a gravitational field on the course of natural processes"

- http://www.bartleby.com/173/22.html

Harald

 

[Fredrik]

Do you guys agree with the above description of how to couple spacetime differential geometry with actual objects?

Most of that sounds good. This one is wrong: "no other theory has the field equations dictating how particles move". (The guy who wrote that probably knows GR, but was sloppy with this statement). Classical electrodynamics has field equations (Maxwell's) that tell us how particles move. What makes GR special is that the most important field is part of what defines a "spacetime". A spacetime isn't a manifold, it's a pair (M,g) where M is a manifold and g is a metric. The metric is a special kind of tensor field on M. The electromagnetic field is another, but it's not considered part of the spacetime structure. Instead it contributes to the stress-energy tensor, which has a relationship with the metric described by Einstein's equations.

Compare with special relativistic classical electrodynamics. Here we have a specific spacetime (M,η). The metric is always η, and the electromagnetic field is just a tensor field on this particular spacetime. Nothing can change the metric in this theory.

In both SR and GR, the electromagnetic field is what determines how charged particles move, in particular how their motion deviates from geodesic motion. In SR, the motion of charged particles will change the electromagnetic field, but not the metric, so the geodesics will remain the same. In GR, the change in the electromagnetic field induces a change in the metric as well, so the geodesics do not remain the same. (However, it would take an absurdly strong electromagnetic field to change the geodesics noticeably).

Everything I mention seem intuitive and can explain General Relativity isn't it? Did not what I mentioned explained it clearly Fredrik? You said "I still haven't found a really satisfactory answer". Is not everything I mentioned above a satisfactory answer?

Those quotes aren't precise enough for me. It's great that they have informed you that that motion is represented by curves in spacetime and that the motion of a test particle in free fall is represented by a geodesic, but they don't make it perfectly clear what the theory says about results of experiments.

 

[bobc2]

Example:

We have a wonderful model of a 4-dimensional universe populated by 4-dimensional objects. At face value special relativity would seem to imply that those 4-D objects are just that--objects frozen in time so to speak--motionless. Yet, we always speak of observers moving along their world lines at the speed of light. If the objects, including 4-D spaghetti-like bundles of neurons extending millions of miles along the 4th dimension, are not actually moving as 4-D objects, then what is doing the moving? And what is the physical significance of the imaginary i that is often attached to the 4th dimension? And where are all of the observers really located at a particular instant of time (they don't share the same simultaneous 3-D cross-sections of the 4-D universe)? Is there some universal synchronized time for all consciousnesses? Or does a consciousness and "NOW" experience exist at all points along the 4-D world lines for every observer?

It's questions such as these that cause physicists (probably most of them) to consider those 4-D objects as mathematical constructs and not real physical 4-D objects. So, with these kinds of mathematical constructs (and we haven't even thrown Quantum Field Theory into the mix) how does mathematics connect to reality (unless, as someone has noted, you go along with Max Tegmark)?

That's why I suggested that at some point along the pursuit of reality rogerl may not find the physics forum a satisfactory place to continue his pursuit. Pursue it here for awhile, yes--but eventually the discussion is not appropriate for this forum (check the rules).