Are Tidal Forces Curvature of Space?

[Grasshopper]

TL;DR Summary

I want to knock this question out once and for all.

I've heard it and I've read* it before, so I just want to make sure I understand this so I never have to wonder about it again.

So, are tidal forces exactly curvature of space?

Here's why I think the answer to that is yes:

.I've seen a spacetime interval equation which has a coefficient on the time term, and a coefficient on the space terms. Further, I'm told* that if you ignore the coefficient on the space components (or rather, set them to 1), it gives an equation that produces the results of Newton's model. Specifically, the equation (with the spatial coefficient set to 1) is this (unless I'm mistaken):

$Δs^2 = (1 - \frac{2GM}{c^2r})(cΔt)^2 - (Δx)^2 - (Δy)^2 - (Δz)^2$

and I have read that the coefficient here shows how time would be curved.

I interpret the equation as this: As r decreases, the entire time term grows (the bigger r gets, the smaller $\frac{2GM}{c^2r}$ gets, so the larger the coefficient is). Intuitively, that means to me that the "tick marks" on time are not evenly spaced as a function of r. They "crunch" as if being "curved."

However, I am told when the coefficient on the space terms is not 1, space curves as well:

$Δs^2 = (1 - \frac{2GM}{c^2r})(cΔt)^2 - (1 + \frac{2GM}{c^2r})[(Δx)^2 + (Δy)^2 + (Δz)^2]$

However, the spatial term is missing a factor of $c^2$, which I would interpret as a reason that tidal forces are small (at least here on Earth).Now, please feel free to eviscerate all this intuitive interpretation if it's way off. I would truly appreciate that, because the goal here is to learn. But whatever parts are close to right, please let me know as well.But most importantly, are tidal forces curved space?One final thought: Another interpretation I'm thinking about is that the entire equation relates to tidal forces, both the space and time components. That also makes sense to me because if the equation with the space coefficients set to 1 approximates Newton, then the time component must also include tidal forces, because presumably Newton includes tidal forces. So in that case, spacetime curvature is tidal forces, not just space curvature.Thanks as always!.
.
*"Read" in particular refers to the source below, and "I am told" here means things I've seen on various physics forums and the rest of the internet, the place where you definitely shouldn't be trying to learn physics, which is why I'm here trying to see if it's right. The only forum I trust is this one. One particular source is from Brown University:

https://www.math.brown.edu/tbanchof/STG/ma8/papers/dstanke/Project/Relativity.html

"When he returned to the problem, he focused on a gap in his previous reasoning: he had ignored tidal forces. Tidal forces are the name given to the forces that result from differences in the strength of the gravitational forces on an object. He had ignored these forces in his earlier work, but their existence invalidated his theory because they allow someone in free-fall to observe the effects of gravity; if you observe your body to be stretching to enormous length, you can be sure you're being pulled by gravity and not floating in empty space. How could Einstein explain tidal forces? The answer lies in curved space."

(bold is my emphasis)

Given my last adventure with posting something from what ostensibly is a reputable source [a QM article from a Journal of Physics: Conference Series], I am not fully trusting it until the experts here confirm or reject it.

 

[pervect]

Tidal forces are curvature of space-time, rather than the curvature of space. The line element for ds^2 you posted gives you the metric, but that's not the same as the curvature, though the curvature can be computed from the line element through a process that involves taking a rather complicated set of second derivatives of the metric components. To oversimplify greatly, think of a function of a single variable. If the variable is constant, there is no curvature. If the first derivative of the function vanishes, there is still no curvature. Curvature appears only when the second derivative is nonzero.

The result of the computation of the space-time curvature is known as the Riemann curvature tensor.

Tidal forces are actually a subset of the Riemann tensor, but they are one of the most important parts. They're also called the electric part of the Riemann. The other two parts are the magnetic part of the Riemann, which is vaguely similar to the magnetic field in electromagnetism, and the topogravitic part, which gives the part of the curvature that's purely spatial (as seen by a specific observer). Rather similar to electromagnetism, an observer who is at rest relative to a static field will see the magnetic part as zero, and an observer who is moving relative to a static field will see a non-zero magnetic part. Thus the breakdown of the Riemann curvature into the three parts depends on specifying an external vector field that represents "an observer". in addition to the tensor itself.

 

[PeterDonis]

are tidal forces exactly curvature of space?

No. They are curvature of spacetime.

I've seen a spacetime interval equation which has a coefficient on the time term, and a coefficient on the space terms.

All such equations are coordinate-dependent. You cannot draw physical conclusions from a coordinate-dependent expression. You need to be looking at invariants; for tidal forces, the relevant invariants are those associated with the Riemann tensor and various contractions of it with other vectors and tensors.

 

[Grasshopper]

Thanks for the detailed responses. They have revealed a lot.
Side note: I’m thinking I have some terrible luck on internet learning, because the Brown.edu page said curvature of space (multiple times). Why would they do that? Brown is supposed to be in the Ivy League for crying out loud!

(To the author’s credit, at one point he does mention “four-dimensional,” but in several instances this qualification is completely left off)EDIT — I believe I have found the answer to my own question. This appears to be from a course about math in four or more dimensions, so it would have been implied.

https://www.math.brown.edu/tbanchof/STG/ma8/

Thank goodness...

 

[Ibix]

I believe I have found the answer to my own question. This appears to be from a course about math in four or more dimensions, so it would have been implied.

Quite probably, yes. Spacetime is a 4d pseudo-Riemannian manifold, which is a space in the mathematical sense.

Do be aware that curved space (in the physics sense of space) is also a thing. It's less important because you can always define space in multiple ways and it has nothing to do with tidal forces, but you will come across the concept, particularly in cosmology where the three FLRW solutions are usually classified by their spatial curvature (+ve, 0, -ve).

 

[vanhees71]

Well, be careful when mathematicians teach physics (and physicists mathematics). The language can be pretty different though they use the same words.

 

[Grasshopper]

Quite probably, yes. Spacetime is a 4d pseudo-Riemannian manifold, which is a space in the mathematical sense.

Do be aware that curved space (in the physics sense of space) is also a thing. It's less important because you can always define space in multiple ways and it has nothing to do with tidal forces, but you will come across the concept, particularly in cosmology where the three FLRW solutions are usually classified by their spatial curvature (+ve, 0, -ve).

Of course if spacetime is curved, then space must also be curved, just in the sense that it's part of spacetime, right? Or is it folly to try to break it down that way? (I mean after all, what is "space" and what is "time" even in special relativity appears to be coordinate dependent, so I would imagine the same is true in GR)

.Regarding FLRW, would these solutions be the ones that determine if the overall geometry of the universe is flat, positively or negatively curved? I ask because you said there are three solutions, so this is a pretty wild stab in the dark. Also, pop physics has taught me (probably erroneously) that the cosmological constant can be interpreted as negative pressure, and that this is related to how the universe expands, if it will expand forever, and why it's accelerating (depending upon some factors related to +, -, or flat space curvature).
.

Pop physics...

Pop physics kind of says that time curvature is the most important cause of "gravity" here on Earth, and "space" curvature yields the little differences from Newton's gravity. However, I suspect this is at best partially accurate, and if I recall correctly, I've been told on this forum not to take watered down interpretation that too seriously.I'm sadly a ways from being able to grasp this stuff. Presently looking through Special Relativity (Woodhouse) published by Springer. It's a bit of a mathematical stretch for me. But math doesn't teach itself.

In the meantime I'll have to continue making obscene threads like this. Apologies in advance. (sidenote: I hope you guys get paid. I can't imagine how annoying it is to deal with this stuff day in and day out, especially when you get a crank in here who thinks he knows the topic of your life's work better than you)

 

[Ibix]

Of course if spacetime is curved, then space must also be curved, just in the sense that it's part of spacetime, right?

No. One of the FLRW spacetimes has flat space. I don't know if it's always possible to find flat spacelike slices, but it's certainly possible in some cases of curved spacetime. And you can easily have curved space in flat spacetime.

Regarding FLRW, would these solutions be the ones that determine if the overall geometry of the universe is flat, positively or negatively curved?

The three solutions have flat, positive and negative curvature of their spatial planes, defining "space" as the set of events where co-moving observers' proper times are equal. The only flat spacetime is Minkowski spacetime (although you can find funky coordinate systems that disguise its flatness). The cosmological constant doesn't change the sign of the curvature, but changes the way the amount of curvature evolves.

Pop physics kind of says that time curvature is the most important cause of "gravity" here on Earth, and "space" curvature yields the little differences from Newton's gravity. However, I suspect this is at best partially accurate, and if I recall correctly, I've been told on this forum not to take watered down interpretation that too seriously.

Yes to the latter. The point is that if you approximate that everything is moving slowly and all gravitational fields are weak then the Einstein field equations (a 4x4 symmetric tensor equation, so ten independent equations) all reduce to 0=0, except for the $tt$ component which reduces to Newtonian gravity in the form of Poisson's equation. Tightening up the approximation again brings in other components like $rr$ or whatever. People get scared by tensor equations, so you see that kind of thing loosely stated as "time curvature is the important thing for our everyday experience of gravity", but that isn't a particularly well-defined thing to say. The bit about the tensor components is the exact truth.

I'm sadly a ways from being able to grasp this stuff. Presently looking through Special Relativity (Woodhouse) published by Springer. It's a bit of a mathematical stretch for me. But math doesn't teach itself.

Don't know that one. I do recommend reading a few textbooks and seeing which ones you like - sometimes somebody else's explanation of something is all you need. I'm fond of Taylor and Wheeler's Spacetime Physics, which is now a free download from E F Taylor's website, and Ben Crowell (@bcrowell) wrote a text that's free to download from www.lightandmatter.com/books.html (his Relativity for Poets is worth a read even if you don't go for his SR book - it's non-mathematical, but about as solid as it can be without maths).

I hope you guys get paid.

No, we don't. You can petition Greg on our behalf if you like...

I can't imagine how annoying it is to deal with this stuff day in and day out

Teaching is fun (as long as there's no marking involved...), but people do leave when they've had enough. The people with a genuine desire to learn more than make up for the occasional nutcase, IMO. And I've learned a lot here - helping others as others help me seems like part of the deal...

 

[Grasshopper]

My preference in textbooks have always been ones that have answers to questions in the back. Unfortunately, that isn't so prevalent in upper division stuff, from what I've seen — especially in math (and understandably so, since it very quickly becomes nothing but proofs). I really like Boas for mathematical methods of physics, though. Other than it being a bit compact (which is to be expected), I have no complaints.

 

[Ibix]

Taylor and Wheeler has answers to odd-numbered exercises, and a quick Google may well turn up answers. If you're really stuck, ask us.