Fields and relativity - a broad question

[Nereid]

"fields" and relativity - a broad question

... 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 http://scienceworld.wolfram.com/physics/GravitationalField.html" 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 textbook-style 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?

 

[pervect]

In the narrow technical sense I think of a field as being a tensor field. The source "fields" of gravitation which contribute to the stress-energy 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 modeled 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 non-zero 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 space-time as the "gravitational field", though this isn't terribly common. This is a true tensor, so it could be subsumed under defintion 1.

 

[Entropia]

It is clear that the term "fields" holds different meanings and implications in different contexts, particularly in relation to relativity. In quantum field theory, fields are used to describe fundamental particles and their interactions, while in general relativity, fields are used to describe the curvature of spacetime caused by massive objects. While both theories involve the concept of fields, they are fundamentally different in their approach and mathematical structure.

In a more general sense, the term "field" is often used in textbooks to explain the concept of a gravitational field, which is a way of visualizing the effects of gravity on objects in space. However, the use of this term can vary depending on the specific context and level of understanding. For example, some may use it to refer to the force of gravity itself, while others may use it to describe the curvature of spacetime.

In popular science writing, the term "gravitational field" is often used to explain the concept of gravity without using equations or math. This can lead to a more simplified understanding of the concept, but it may also lead to misconceptions or misunderstandings. In a broader sense, the term "field" may also be used in a variety of contexts, such as in science fiction or literature, where it may have different meanings or implications.

Overall, the term "fields" and its relation to relativity is a broad and complex topic that can be interpreted in different ways depending on the context and level of understanding. It is important to consider the specific context and implications of the term when discussing it in relation to relativity.