The physical nature of spacetime

[MeJennifer]

It's somewhat ironic that some of the key objects of study in General Relativity - like the metric and the Riemann - are in fact absolute in character.

I beg to differ.

The metric of space-time completely depends on the influence that each particle has on space-time in a non linear way.

When I look at what otherwise appears to be empty space, I am more confident now that the space is not empty but alive with the metric tensor field of GR. I can test that it exists by placing test particles there and seeing the effect of the field on the particles.

But in reality there are no test particles and if you interject a particle into the "soup", the metric will change!

This element to me is the essense of the theory of relativity.

For instance take the Schwarzschild solution. Is it an exact solution? Yes, if there are no other particles!
As soon as we have one other particle the solution is not longer exact but at best an approximation.

 

[coalquay404]

I wish everybody would stop calling space-time curved in cases where the Riemann tensor is zero, it gives great confusion and a lot of wasted time in explaining.

Simply because a physical observable has value zero does not mean that it's not an observable.

 

[jimbobjames]

@MeJennifer: "The metric of space-time completely depends on the influence that each particle has on space-time in a non linear way."

I was referring to absolute in the sense that the metric and Riemann are geometric objects which are independent of any choice of reference frame.

And yes, you are right - even test particles with negligable mass will still have an effect on the field, however vanishingly small that effect may be.

 

[masudr]

But in reality there are no test particles and if you interject a particle into the "soup", the metric will change!

This element to me is the essense of the theory of relativity.

We use test particles in electrodynamics too! Even in non-relativistic field theories. We know that there is no such thing: test particles are defined by a limiting process in order to help understand nature better. No one is claiming a test particle will not affect the field in question (whether it be the gravitational, Newton gravitational, electromagnetic etc.)

 

[MeJennifer]

No one is claiming a test particle will not affect the field in question (whether it be the gravitational, Newton gravitational, electromagnetic etc.)

And I was not implying that.

 

[MeJennifer]

I was referring to absolute in the sense that the metric and Riemann are geometric objects which are independent of any choice of reference frame.

Of course they are, they are indeed mathematical objects, e.g. abstractions.

General relativity is a relational theory, the manifold has no independent existence, it simply represents the relationships of all space-time events. Or as Rovelli wrote: "We can say that GR is the discovery that there is no spacetime at all".

Note that it is impossible to separate any kind of background on which events happen in general relativity!

With regards to the manifold, see for instance what Rovelli writes in Quantum Gravity:

" In the mathematical formalism of GR we utilize the "spacetime" manifold M, coordinated by x. However, a state of the universe does not correspond to a configuration of fields on M. It corresponds to an equivalence class of field configurations under active diffeomorphisms. An active diffeomorphism changes the localization of the field on M by dragging it around. Therefore localization on M is just gauge: it is physically irrelevant.
In fact, M itself has no physical interpretation, it is just a mathematical device, a gauge artifact. Pre-general-relativistic coordinates x^u design points of the physical spacetime manifold "where" things happen...; in GR there is nothing of the sort. M cannot be interpreted as the "set" of physical events, or physical spacetime points "where" the fields take value. It is meaningless to ask whether or not the gravitational field is flat around the point A of M, because there is no physical entity "spacetime point A". Contrary to Newton and Minkowski, there are no spacetime points at all. The Newtonian notions of space and time have disappeared. "

Pretty profound right?

 

[jimbobjames]

@MeJennifer: Thanks a lot for pointing me to these ideas from Rovelli.

From the piece you quoted however, I see no contradiction with what I was saying above.

"localization on M is just gauge: it is physically irrelevant"

and

"M itself has no physical interpretation, it is just a mathematical device"

Above I was coming to the conclusion that the physically significant (ie very real) object was the field (modelled mathematically by the metric and its derivatives) and not necessarily the manifold.

The very real existence of that field could be demonstrated by the instantaneous effect it would have on test particles, which you might introduce into the field, even at large spatial separations from the source of the field.

Were there nothing real (physical) at the location of the experiment - which would otherwise appear to be "empty" space - then there would be no explanation for the relative acceleration of the test particle along a line directly to the centre of mass of the source (and not in any other arbitrary direction).

(I admit that I have only carried out the experiment myself at much shorter distances however )