Talk:Curved spacetime

Why the split from Spacetime?

My main objection to the split is that authorship has been completely lost. In 2017, under my former user name of Stigmatella aurantiaca, I wrote almost the entirety of this section's text. Now there is no clue that I had anything to do with its writing. As a result of your splitting out of this section, the Authorship page for Spacetime, which previously showed 85% authorship under my former and current user names, now shows only about 70% authorship.

I admit to being motivated by a bit of vanity. I like knowing that after I'm gone in just a few years, a major part of what readers continue to learn about Special relativity and Spacetime will be from material that I've written, even though my identity is hidden. Prokaryotic Caspase Homolog (talk) 16:43, 7 September 2024 (UTC)

The loss of authorship is definitely frustrating. If you read the page Wikipedia:Splitting it will likely give you some idea about why a split was done here. I didn't see a vote or discussion on the matter, but would have voted in favor if I did. The page Spacetime currently has 10,955 words. The new page Curved spacetime is 3,115 words. According to the general guidelines about article length on the page for splitting, articles around 9,000 words "Probably should be divided or trimmed, although the scope of a topic can sometimes justify the added reading material," and articles around 15,000 words "Almost certainly should be divided or trimmed." Unfortunately, the page for spacetime was just too long, and as it sits now could probably be edited to be a bit shorter. It sucks that you're authorship statistics are not accurately represented, but 70% is still ALOT of text. I'll add a post on this talk page that mentions the split for posterity. Edit histories are recorded, anyone interested can still look back and see what was written by who. GeogSage ^{(⚔Chat?⚔)} 03:13, 9 September 2024 (UTC)

Does "cuvature of time" have any sense?

In think, no. Time is a one dimensional manifold. One dimensional manifolds are flat (they have no curvature). I'm afraid the author mismatches time dilatation with curvature. Spacetime can have curvature, space can have curvature, but time can't. The referenced paper also doesn't contain such thing as "time curvature". 'n Quijote (talk) 18:13, 7 September 2025 (UTC)

I replaced "curvature" to "distortion" everywhere where it was used incorrectly.

'n Quijote (talk) 06:34, 9 September 2025 (UTC)

@'n Quijote: The expression "curvature of time" is found throughout the internet, including Sean Carroll's blog:

https://www.preposterousuniverse.com/blog/category/arxiv/page/5/

Sean Carroll is as authoritative a source as you can get. The pedagogical point is that in the Newtonian (weak-field, slow-motion) limit, gravitational effects can be attributed almost entirely to the time-component of the metric.

The expression "curvature of time" is physically meaningful and widely used pedagogically.

In a strict sense, you are correct in that the Riemann curvature tensor cannot be decomposed cleanly into "curvature of space" + "curvature of time" and is not a precise invariant concept in GR.

But your replacement of all instances of "curvature" with "distortion" is also incorrect. It does not map cleanly onto:

• the metric
• curvature tensors
• or any other standard GR object

In the interests of pedagogy, I had perhaps overstated “curvature of time,” but your edits overcorrect by replacing standard GR language ("curvature") with less precise wording ("distortion").

Would you accept the following compromise wording?

In general relativity, gravity is described by the curvature of spacetime. In the weak-field limit, many gravitational effects can be understood primarily in terms of variations in the flow of time (gravitational time dilation), sometimes informally described as "curvature of time," although this is not a precise geometric notion

"Curvature of time" is legitimate pedagogy, and "distortion everywhere" is not an upgrade. The ideal fix would have been to qualify the original wording, not the universal replacement which you effected.

Prokaryotic Caspase Homolog (talk) 02:44, 12 May 2026 (UTC)

No, it was not replaced "everywhere", only when it was used incorrectly (i.e., in a sense that differed from its definition). I don’t think it’s a good idea to use the same word with two different meanings in a single article (it's not pedagogical, it's confusing). Do you want to explain whether it’s the literal or metaphorical usage every time it appears? I would rather claarify the meaning of "distortion": it is the change of metric (some change causes curvature, others do not). 'n Quijote (talk) 20:03, 12 May 2026 (UTC)

Nevertheless, I can find multiple instances of the use of the expression "curvature of time" across the internet (such as in Carroll's blog), and in semi-popular books such as "Gravity from the Ground Up" by Schutz (see excerpt https://www.gravityfromthegroundup.org/pdf/timecurves.pdf). Schutz, incidentally, never uses the expression in his serious textbooks written for physics students, but as you can see from my excerpt, he does believe usage of the expression is justified in his semi-popular writings. Given that two highly respected physicists have given their imprimatur to use of the expression in semi-popular contexts, I believe that use of the expression in a Wikipedia article is justified. I do not believe that substituting with the unclear term "distorted" is correct. Prokaryotic Caspase Homolog (talk) 08:10, 13 May 2026 (UTC)

I don’t understand why you say the term “distorted” is unclear. It has a precise meaning: an altered metric structure. This is entirely consistent with the word’s everyday meaning. Why would the term “curved” be any clearer? Schutz puts it this way:

Temporal curvature is nothing more than gravitational redshift: time moves forward at different speeds in different places, so time is curved.

Sorry to say, but if I want to use the word “curved,” then I have to say that “spacetime is curved” (space is also involved in the explanation above!). When I say that something is curved, it means that the parallel transport of a tangent vector along a closed loop results in a change in the vector’s direction. But if I transport a tangent vector only in the direction of time along a closed loop, it does not result in a change. So time is not curved, no matter who claims otherwise. 'n Quijote (talk) 14:42, 13 May 2026 (UTC)

You are correct that "curvature" has a precise, invariant meaning in general relativity, being encoded in how parallel transport around a loop changes a vector, as expressed by the Riemann curvature tensor $R_{\sigma \mu \nu }^{\rho }$ which measures the failure of covariant derivatives to commute: $(\nabla _{\mu }\nabla _{\nu }-\nabla _{\nu }\nabla _{\mu })V^{\rho }=R_{\sigma \mu \nu }^{\rho }V^{\sigma }$.

In contrast, "distortion" does not relate to a single standard geometric object. Doing a literature search, I do see informal use of the word "distortion" to several distinct effects that do have precise meanings. I see "distortion" loosely used in discussions of tidal effects (Weyl curvature), the distortion of congruences (optical scalars), in discussions of black holes influenced by external gravitation fields (so that their geometries differ from the isolated Schwarzschild/Kerr solutions), and to refer to coordinate distortion where a shape looks warped due to projection or embedding.

So your objection is to

1. what would technically be misuse in a non-technical context of the word "curvature" that has a precisely defined meaning in general relativity,
2. preferring to replace it with the word "distortion" that is not a fundamental invariant concept in GR, but which is informally used in various contexts to refer to a variety of distinctly different effects, none of which correspond to non-commutivity of covariant derivatives?

I really don't understand why you would prefer "distorted time" versus "curved time" in the context of a discussion that I had originally written some ten or so years ago (?) that was aimed at a popular level, especially when I see the expression "curvature of time" used in semi-popular settings by people whom I would consider experts. In particular, I had modelled my discussion specifically after Schutz's presentation in his semi-popular work "Gravity from the Ground Up". Prokaryotic Caspase Homolog (talk) 18:18, 13 May 2026 (UTC)

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I believe that I may have come up with a workable compromise. A well-known quote, often attributed to Larry Davis, is that "a good compromise is when both parties are dissatisfied". Although in this edit, I continue to use the expression "curvature of time", I make sure that in the first instance that the reader encounters this expression (in the section title), the expression is bracketed by quote marks. The second instance is also bracketed by quote marks and is followed by the brief disclaimer, with a note pointing to a detailed explanation.

You clearly know the material at an advanced level, and I can appreciate how you might be offended by the lack of precision in this expression. For my part, I dislike apologizing to the reader in an article aimed to a popular audience for committing a verbal misdemeanor that the typical reader would have no means of appreciating even with careful explanation. For me, the compensatory benefit is that using equivalent language for space and time places the two on equal footing.

I believe that my current edit should give us both a roughly equal amount of dissatisfaction. Truce? Prokaryotic Caspase Homolog (talk) 01:00, 16 May 2026 (UTC)

This must be a very good compromise, since I am very dissatisfied. But I don't want to fight; let others judge this issue. 'n Quijote (talk) 07:23, 16 May 2026 (UTC)

I'm genuinely sorry that I couldn't work out a better compromise. Prokaryotic Caspase Homolog (talk) 16:25, 16 May 2026 (UTC)

We have been fighting two separate battles here:

1. a technical battle over what "curvature" means in differential geometry, and
2. a pedagogical battle over how to communicate weak-field GR intuition to non-specialists.

These are not the same dispute.

Thinking over my last edits, I decided to change Newtonian gravity as the "curvature of time" to Newtonian gravity and gravitational time dilation. Right now, I'm at a bit of a loss as how better to tilt the article to appease your unhappiness. I absolutely do not like the idea of replacing multiple instances of the word "curvature" with "distortion", which lacks a canonical meaning. Prokaryotic Caspase Homolog (talk) 17:18, 16 May 2026 (UTC)

Much better. Thanks! 'n Quijote (talk) 18:55, 16 May 2026 (UTC)

You're welcome!

This article was a section split out from Spacetime about a year and a half ago. I originally wrote most of the material in this article about 10 years or so ago. I have no responsibility for the lead paragraph, but most of the rest feels like it still remains about 90% my writing.

You've forced me to re-read the article carefully. Upon re-reading, I note that the article's coverage is a bit idiosyncratic. When I wrote it, I wanted to cover different topics than covered in the General relativity article and using a more consistent tone. I wanted the article to be about the conceptual geometry of GR but written at a semi-popular level rather than being some sort of compressed duplicate of the General relativity article.

Given our dialog, it appears that a major defect of the article as it stands is that it never quite explains what curvature is geometrically in a coordinate-independent way before using it operationally throughout. I discuss geodesics, tidal effects, Christoffel symbols, metric coefficients, gravitomagnetism etc., but the reader never gets a clean conceptual bridge from flat Minkowski spacetime to genuinely curved Lorentzian manifolds.

So maybe this article needs a whole introductory section on curvature, maybe covering topics like

• Intrinsic curvature and geodesic deviation
• Curvature versus coordinates
• Embedding diagrams and misconceptions
• Possibly a short section on local flatness and normal coordinates

Thoughts? Prokaryotic Caspase Homolog (talk) 20:22, 16 May 2026 (UTC)

I greatly appreciate the heroic work you have done here. I welcome the ideas you have just presented. My knowledge of general relativity is very basic, but I know that spacetime is an observer-independent entity, while gravity is observer-dependent. Curvature is a property of spacetime (and thus observer-independent), while gravity is a property of its coordinatization, i.e., of the observers. An accelerating reference frame does not change the metric of spacetime (only its expression in coordinates), so it cannot create curvature; still it does create gravity even in a completely flat Minkowski spacetime. The program you outlined might clarify these issues. 'n Quijote (talk) 07:10, 17 May 2026 (UTC)

Need to use "curvature" properly

All along the article, the word "curvature" is used as a synonym for geometry, but geometry and curvature are two different concepts. 'n Quijote (talk) 06:33, 9 September 2025 (UTC)

What curvature means

I note that the concept of curvature can be exactly explained, without any formula, as follows.

The mathematical model of spacetime is a pseudo-Riemannian manifold. Pseudo-Riemannian manifolds are differentiable manifolds endowed with a pseudo-metric. On the one hand, this pseudo-metric determines the metric geodesics, i.e., curves of extreme length connecting two points. On the other hand, it also determines the parallel transport (parallel shift) of tangent vectors along arbitrary curves. The concepts of parallel transport and geodesics themselves don't require a metric; they can be defined on an arbitrary differentiable manifold. The rule of parallel transport of tangent vectors of a differentiable manifold is called a connection on its tangent bundle. Once a connection is given, geodesics are defined as such curves that have the property that their tangent vectors at different points are parallel shifts of each other.

The concept of curvature can also be defined on an arbitrary connection, not only one that comes from some metric or pseudo-metric, and this definition applies to pseudo-Riemannian manifolds (hence on spacetime too) as well. Loosely speaking, a connection is curvature-free if the result of a parallel transport from one point to another is independent of the curve along which the tangent vector is transported between the two points. On a simply-connected manifold, this definition is exact, while on multiple-connected manifolds, it defines only the trivial holonomy, not vanishing curvature.

The standard example of the curvature of a connection is the curvature of sphere and as it is demonstrated on this picture, taken from here.

Parallel transport (of the black arrow) on a sphere. Blue and red arrows represent parallel transports in different directions but ending at the same lower right point. The fact that they end up pointing in different directions is a result of the curvature of the sphere.

Based on the concept of curvature described above, the curvature of spacetime can theoretically be demonstrated with the following experiment. Consider two identical gyroscopes placed close to each other and rotating with parallel axes of rotation. Move the gyroscopes along different paths so that they approach another common point. If spacetime is curved, the axes of rotation will not be parallel at the end.

But unfortunately, I have no idea how to neatly incorporate this into the article. 'n Quijote (talk) 09:54, 19 May 2026 (UTC)

I can think of at least three places which might need an introductory treatment of this sort:

• Gravity Probe B as part of a theory section on the geodetic effect, where you should also include a description of Lense–Thirring precession.
• Geodetic effect which is not useful for a person wanting to learn the subject for the first time. The existing explanation in this article is geared for people who are already familiar with the math, which doesn't make too much sense to me.
• Parallel transport which, IMHO, is also not useful for a person wanting to learn the subject for the first time. A person needing help getting through his GR textbook would be sadly disappointed if he/she expects the Wikipedia article to assist him/her in understanding the material.

Common defects of most technical articles in Wikipedia are that

• Contributors add local improvements,
• Contributors do not consider the requirements of effective pedagogy,
• Nobody owns the global structure.

Prokaryotic Caspase Homolog (talk) 18:08, 19 May 2026 (UTC)

---

Another thing that you might consider doing is to add an Appendix to the article for miscellaneous material that doesn't really fit in with the main narrative.

Consider, for instance, the Spacetime#Technical_topics section. When I was reorganizing this article back in 2017, I needed a place where I could shove off-topic material that was already present in the article, and where I could stuff in new off-topic material of my own (Is spacetime really curved? is my writing).

Prokaryotic Caspase Homolog (talk) 18:24, 19 May 2026 (UTC)

@'n Quijote: I took the liberty of preparing your explanation as an Appendix to the current article. In addition to adding references, I made a few minor editorial revisions.

• "extreme length" ==> "extremal length"
• removed "parallel shift" (synonym clutter)
• simplified "such curves that have the property that"
• reworded "from one point to another"
• "curvature of sphere" ==> "surface of a sphere"
• "approach another common point" ==> "later meet again at a common point"
• "will not be parallel at the end" ==> "will generally no longer be parallel" (you mentioned that some weird stuff can occur in flat multiply connected spaces)

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Appendix

Parallel transport and spacetime curvature

Parallel transport (of the black arrow) on a sphere. Blue and red arrows represent parallel transports in different directions but ending at the same lower right point. The fact that they end up pointing in different directions is a result of the curvature of the sphere.

The mathematical model of spacetime is a pseudo-Riemannian manifold. Pseudo-Riemannian manifolds are differentiable manifolds endowed with a pseudo-metric. On the one hand, this pseudo-metric determines the metric geodesics, i.e., curves of extremal length connecting two points. On the other hand, it determines the parallel transport of tangent vectors along arbitrary curves.

The concepts of parallel transport and geodesics do not themselves require a metric; they can be defined on an arbitrary differentiable manifold. The rule governing the parallel transport of tangent vectors on a differentiable manifold is called a connection on its tangent bundle. Once a connection is given, geodesics may be defined as curves whose tangent vectors remain parallel to themselves under parallel transport.

Curvature can likewise be defined for an arbitrary connection, not only for one derived from a metric or pseudo-metric. Loosely speaking, a connection is curvature-free if the result of parallel transport between two points is independent of the path along which the vector is transported. On simply connected manifolds, this condition is equivalent to vanishing curvature. On multiply connected manifolds, however, it characterizes only trivial holonomy.

A standard example of curvature arising from parallel transport is the surface of a sphere, as illustrated in the accompanying figure.

Based on the concept of curvature described above, spacetime curvature could theoretically be demonstrated by the following experiment. Consider two identical gyroscopes placed close to one another with parallel axes of rotation. Move the gyroscopes along different paths so that they later meet again at a common point. If spacetime is curved, the axes of rotation will generally no longer be parallel.

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Please correct any goofs that I made in my editorial revisions and add as an appendix. You will need to fix the duplicate citation of Carroll, which is already cited in the article. Prokaryotic Caspase Homolog (talk) 07:48, 21 May 2026 (UTC)

I didn’t intend this text for the article itself; I wrote it for you to consider, so that you might be able to incorporate this basic definition of curvature into the article in a meaningful way, since I believe it’s missing from there. But as it stands, in the form of an appendix, it would feel a bit awkward. But do as you see fit. 'n Quijote (talk) 05:09, 22 May 2026 (UTC)

If I add the section, authorship will be attributed to me, not to you. I'm a little sensitive about authorship. For example, the way this article was split out of Spacetime, User:Fgnievinski gets credited with 78% authorship when in reality, most of the writing (except for the lead paragraph) was my work. https://xtools.wmcloud.org/authorship/en.wikipedia.org/Curved%20spacetime/

I have no desire to take credit that belongs to you. You've written a valuable introduction to the concept of parallel transport which is far more accessible than the Parallel transport article. Prokaryotic Caspase Homolog (talk) 05:48, 22 May 2026 (UTC)

Let's imagine a beginning student of GR who is struggling through, say, Shutz's textbook. His Chapter 6.4 "Parallel-transport, geodesics and curvature" is clearly written, but let's suppose the student has a mental block and thinks consulting Wikipedia might help. So the student consults the Wikipedia Parallel transport article hoping to find a good starting explanation with a different point of view. Does he/she get help from this article? Absolutely not! This sort of problem has been around since Wikipedia's beginnings. Back in 2006, Chris Hillman quoted:

...I have noticed that some scientific and technical articles are being edited to prevent other users from obtaining a complete and full comprehension of a topic in the same manner as a member of a trade or artisan guild might try to hide techniques or methods or understanding of what the topic actually involves. Such articles are only permitted to have a highly technical version or explanation of the process being presented in the same manner as a tradesman or artisan might withhold simple explanations from a patron for the sole purpose of mystifying the topic and keeping the patron from knowing “too much.”

I think your contribution is absolutely the sort of thing that Wikipedia needs more of, and I would be glad to see it appended to this article, or somehow or other wormed into Parallel transport (an almost impossible task, given the latter article's structure). Prokaryotic Caspase Homolog (talk) 06:45, 22 May 2026 (UTC)

Thank you for your thought-provoking comments. I'll try to figure something out, but I need some time to do so. 'n Quijote (talk) 08:29, 22 May 2026 (UTC)

OK, take your time. Whether you add your section as an Appendix to this article (written at a somewhat higher level than the current main text) or revise it so that it serves as an intuitive introductory exposition to the Parallel transport article doesn't really matter so long as it does get into Wikipedia somewhere. I do not have the skills or background to add an appropriate introductory section to Parallel transport, but I sense that you do.

I have absolutely no idea of the target audience for Parallel transport. In its current form, it is not something that my hypothetical junior/senior-year GR student needing an alternative presentation from the textbook could learn anything from. Prokaryotic Caspase Homolog (talk) 09:22, 23 May 2026 (UTC)

I added a revised form of my above comments to the talk page for Parallel Transport Prokaryotic Caspase Homolog (talk) 10:00, 23 May 2026 (UTC)

What is pedagogical?

Pedagogical could mean "easy to understand". But "easy" depends on the existing knowledge of the reader. So, I would rather say something is "pedagogical", if it requires the less prerequisits to understand it. This definition doesn't depend on the existing knowledge of the reader. Curvature as currently explained in the article, requires the knowledge of coordinates, Pythagorean theorem, metric, and so on. You may say that my definition requires much more: differentiable manifolds, fiber bundles, connections, parallel transport, and so on. But this is not true. You can explain to a kindergartener why his ball is curved. You can demonstrate the parallel transport with a pencil slid on the ball, and show the different results sliding on different paths. So In my view, my definition is much more pedagogical than yours. 'n Quijote (talk) 07:13, 21 May 2026 (UTC)