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Fields and relativity  a broad question 
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#1
Jan2206, 05:43 PM

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... being a general enquiry into the uses and meanings of the word "field(s)" in relation to Einstein's theories of Special Relativity and General Relativity.
First, the narrow technical meanings. QFT ('quantum field theory') involves 'fields'. In what sense, if any, does GR (General Relativity) also involve 'fields'. By 'involve' here I mean both the textbook explication of the core equations and relations(hips), as well as the underlying mathematical structure (to as many levels as you wish). To the extent that they both involve 'fields', how similar are these 'fields'? Next, the more general (but still narrow) meanings. In 'textbook' material  such as this from Eric Weisstein's World of Physics (I typed 'gravitational field' into Google and chose the first hit that looked 'textbooky')  the concept of a 'gravitational field' seems alive and flourishing (Google tells me that there are >4 million hits to my simply enquiry; an eyeball estimate of the first half dozen webpages of hits suggests that many of these are from textbookstyle webpages (ignoring crackpot sites, of course)). Do readers of this post have a feel for the most common ways that this expression/concept/term ("gravitational field") is tied to GR  in terms of both the textbook approach/explication, and the core aspects of the theories (which of course include Newtonian gravity)? Finally, the popsci/folk/general meanings. By 'popsci' here I mean popular science writing on the topic of gravitation and relativity; specifically, that which seeks to explain the ideas, concepts and theories without using any equations or math. By 'folk/general' I mean use of the terms outside any of the environments described above, such as on Star Trek, in computer games, literary criticism, etc. This is, of course, a vastly bigger field than all the above combined, yet it is the one in which more Joe Sixpacks and Joan Chardonnays will encounter 'gravitational field'. What opinions do readers of this post have concerning the ranges of meanings that are to be found here? 


#2
Jan2206, 06:08 PM

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In the narrow technical sense I think of a field as being a tensor field. The source "fields" of gravitation which contribute to the stressenergy tensor (like the electromagnetic field) would be such tensor fields. Tensor fields can be defined by the way that they transform. Knowledge of a tensor field at a point in one coordinate system allows one to calculate the tensor field at the same point in any coordinate system, so tensors can be thought of as coordinate independent objects.
In the popular science sense, the "gravitational field" is often taken to be the Christoffel symbols in GR. These are not tensors because of their transformation properties, and because their values are coordinate dependent. As an example of this usage, the "gravitational field" in an accelerating spaceship points in a direction opposite to the acceleration. This "felt" gravity is mathematically perfectly modelled by the Christoffel symbols. This notion is coordinate dependent, because the only difference between a "felt" gravity of zero for a stationary spaceship and a "felt" gravity of nonzero for an accelerating spaceship arises from the motion of the spaceship. Thus the "gravitational field" in this sense depends not only on the point in space, but on one's motion (acceleration) through that point. These are the most popular, IMO, but there are other usages. On occasion, people might refer to the metric of spacetime as the "gravitational field", though this isn't terribly common. This is a true tensor, so it could be subsumed under defintion 1. 


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