I What is the mechanism by which energy/stress curves spacetime

[Sorcerer]

I have seen the equations relating energy, momentum, pressure/stress to the metric tensor and the curvature of spacetime, as I’m sure most people have. But what is the mechanism?

I’ve read an interpretation in Quora (horrid source, I know) where energy and stress are basically “pushing” space or displacing it, causing it to be inhomogeneous in a non-linear fashion, which then leads to a curved spacetime model for plotting world lines and so on.

Is that remotely accurate? Is there another explanation, or is it just one of those “this is the way the universe is and no one knows why” situations?

Thanks. This thread is marked I level, but I welcome responses from every level.

 

[Drakkith]

As far as I am aware, there is no underlying mechanism. It's a fundamental fact that can't be explained in terms of other physical processes. Or, like you said, it's just the way it is.

 

[Ibix]

In GR, that's just what spacetime does in the presence of stress-energy. There is no further explanation than the Einstein Field Equations, which boil down to "the geometry of spacetime depends on the stress-energy content (which is affected in turn by the geometry)".

Beyond that, we have many explanations for how it happens. Fierz-Pauli quantum gravity, string theory, loop quantum gravity, shape dynamics... We just don't know which (if any) is right.

 

[Ibix]

...oh yes, and the Quora explanation is a non-explanation. How is stress-energy supposed to be pushing on spacetime? What forces are involved and how are they transmitted? At least as you've presented it, it's just a pretend-there's-more-to-it version of my "that's what the Einstein Field Equations say".

 

[Sorcerer]

...oh yes, and the Quora explanation is a non-explanation. How is stress-energy supposed to be pushing on spacetime? What forces are involved and how are they transmitted? At least as you've presented it, it's just a pretend-there's-more-to-it version of my "that's what the Einstein Field Equations say".

Here's the link so someone with knowledge can make heads or tails of it.

<unacceptable reference deleted>

 

[PeroK]

Here's the link so someone with knowledge can make heads or tails of it.

You're not going to learn much from a Quora thread, as it includes a lot of nonsense and personal theorising and it's impossible to separate the wheat from the chaff.

And, it's certainly not a valid reference for a scientific discussion of GR.

In physics in general, once you get to the bedrock of a theory there is no "mechanism", as that implies a lower level of the theory. You can explain a phenomenon in terms of underlying concepts (and call these "explanations" or "mechanisms") but when you reach the fundamental concepts of your theory there is, by definition, no more fundamental explanation or mechanism. In GR, the fundamental concept is the relationship between the stress-energy tensor and the curvature of spacetime. And that is as far as physics takes you. Until, possibly, you have a more fundamental theory of quantum gravity, say.

 

[Ibix]

Here's the link so someone with knowledge can make heads or tails of it.

Hm. I think that's worse.

Spacetime is curved, not space, he says. But that's not correct the way he's representing it. The point is that your choice of what slice of spacetime you choose to call space is arbitrary. It can be curved in flat spacetime (that's what an accelerating reference frame is). But you only get gravity if spacetime is curved - which is the distinction the Physics FAQ is making. But he says spacetime is curved, but space isn't curved it's inhomogeneous. Which is a mess. If he wants to call curved spacetime inhomogeneous spacetime I don't have a problem. But inhomogeneity isn't something over and above curvature - it's just a different interpretation of the maths (edit: at least as I understand it - I should probably read the linked paper before being quite so declarative, but its title accords with my understanding).

And changing "curved" to "inhomogeneous" is still not an explanation, even without the confusion. "Stress-energy makes space inhomogeneous" doesn't have any more content than "stress-energy makes space curved". An actual explanation would be a successor theory to GR.

You're not going to learn much from a Quora thread, as it includes a lot of nonsense and personal theorising and it's impossible to separate the wheat from the chaff.

It's easy. @Simon Bridge is a poster here...

 

[Jimmy]

That Quora poster was @John Duffield aka @Farsight. Author of Relativity+, a crackpot physics book. He's long since been banned from physicsforums.

 

[Dale]

But what is the mechanism?

The mechanism is the EFE. That is the purpose of the equations: to describe the mechanism.

 

[Ibix]

That Quora poster was @John Duffield aka @Farsight. Author of Relativity+, a crackpot physics book. He's long since been banned from physicsforums.

And there is the problem with my argument from authority...

I must say I thought I recognised the name, but neither of the aliases autocomplete for me. Curious.

 

[haushofer]

One of the relevant questions here is whether gravity is a fundamental force or emergent. The answer to this question gives insights into whether the "gravity as spacetime curvature" is just a classical picture of something more deep. One example of this is Verlinde's entropic gravity.

But even if gravity is fundamental, the classical picture can still hide what's underneath the hood. E.g., in Fierz-Pauli one can see how the interaction of massless spin-2 particles can make a classical picture emerge of spacetime curvature, something which is not clear at the perturbative level. Something similar happens in the AdS/CFT picture, where superconformal Feynman diagrams have "hidden" (i.e. implict) information about gravity in them.

Personally I think AdS/CFT will give us some insights into the true nature of space, maybe time, and hence gravity. But how exactly I (of course) don't know ;)

 

[Sorcerer]

My issue is that indeed. "Force." When people talk of gravity as a force, I stop and think, but isn't the natural state of motion movement along a geodesic, and the only force experienced is when something IMPEDES that motion? For example, in Newton, a hanging apple has a downward gravity force and an upward tension force (right?), but in GR, there is only one force, the upward force, and as soon as the apple is broken from the branch, it simply follows the natural geodesic path in a way analogous to Newton's law (every inertial object remains in constant inertial motion unless acted on by a force).

 

[Sorcerer]

And there is the problem with my argument from authority...

I must say I thought I recognised the name, but neither of the aliases autocomplete for me. Curious.

Just to be clear, I was not in any sense of the word making ANY argument from authority. I simply posted the link to give you the entire story of what I read in trying to answer this question. That is why I brought it here, so that you guys could explain the flaws in it.

Trust me, not only do I not trust cranks, I despise them with an unholy passion.

 

[Ibix]

Just to be clear, I was not in any sense of the word making ANY argument from authority

I did, though, by arguing you could trust Simon's response because he has an account here while missing that Duffield also had an account. There's a slight difference in posting histories and only one isn't banned, but you need to be a bit more careful than I was being.

 

[Dale]

@Sorcerer back to your question, I am not clear on what qualifies as a “mechanism” in your view. For me, the EFE is the mechanism (or rather it describes the mechanism in the human language most suitable for describing physical mechanisms, math)

 

[Matterwave]

My issue is that indeed. "Force." When people talk of gravity as a force, I stop and think, but isn't the natural state of motion movement along a geodesic, and the only force experienced is when something IMPEDES that motion? For example, in Newton, a hanging apple has a downward gravity force and an upward tension force (right?), but in GR, there is only one force, the upward force, and as soon as the apple is broken from the branch, it simply follows the natural geodesic path in a way analogous to Newton's law (every inertial object remains in constant inertial motion unless acted on by a force).

Yes, this is right. In the Newtonian view, Gravity is a force to be counter-acted in order to remain "stationary". In GR free fall motion is the "natural" force-free state. I'm not sure how this relates back to your original question regarding a "mechanism" by which Stress-Energy curves space time though.

 

[romsofia]

This question is deeper than you expect. When you say that something travels along its' geodesic, are you also considering the self-force that will have it also move off its' wordline?Even in the case of a falling charged massive object, just by existing this will create a current in the world line, causing it move off its' wordline! The curvature in spacetime is so subtle that even just the E+M field created by one charged particle will take it off the path originally in spacetime!

So what is natural isn't even "natural", because of self-force. It's crazy how powerful and weak GR is at the same time. Truly it must be very fundamental! Now for why it actually "curves" spacetime, we may never know. Certain groups think that spacetime must be emergent, and from there you can create a sense of curvature, but that would be closer to quantum gravity, and not classical.

For now, we just admire the genius of others to even think of this beautiful idea...

 

[PeterDonis]

are you also considering the self-force that will have it also move off its' wordline?

A neutral test particle has no "self-force". A geodesic is defined as the path through spacetime of a neutral test particle that is subject to no external forces.

The curvature in spacetime is so subtle that even just the E+M field created by one charged particle will take it off the path originally in spacetime!

This has nothing to do with the curvature of spacetime. An EM force on a charged particle will cause its worldline to not be a geodesic even in flat spacetime.

 

[Sorcerer]

@Sorcerer back to your question, I am not clear on what qualifies as a “mechanism” in your view. For me, the EFE is the mechanism (or rather it describes the mechanism in the human language most suitable for describing physical mechanisms, math)

Well, and this is NOT a personal theory, just an example to show what I hope to find: if you drop a rock into water, it “curves” the water via displacement (and other things relating to the momentum it imparts during the collision and so on). I know spacetime isn’t a fluid, so an exact analogy won’t apply, but some sort of cause is what I’m looking for. Going the other way in GR, that the curvature of spacetime directs the motion of objects along geodesics, does make perfect sense to me, on the other hand.

For what it’s worth, in SR for me an acceptable cause of the relativity of space and time is the invariance of the speed of light, because it’s a logical necessity that if every inertial reference frame measures the same finite maximum speed, they cannot all measure the same time and space (or at least one of the two).

So I’m looking for some sort of mechanism or logical necessity. Could it be relating to something like the equivalence of inertial and gravitational mass? That’s the kind of thing I’m looking for.Of course as you say, the EFEs might be enough if I understand them better.

 

[pervect]

Well, and this is NOT a personal theory, just an example to show what I hope to find: if you drop a rock into water, it “curves” the water via displacement (and other things relating to the momentum it imparts during the collision and so on). I know spacetime isn’t a fluid, so an exact analogy won’t apply, but some sort of cause is what I’m looking for. Going the other way in GR, that the curvature of spacetime directs the motion of objects along geodesics, does make perfect sense to me, on the other hand.

For what it’s worth, in SR for me an acceptable cause of the relativity of space and time is the invariance of the speed of light, because it’s a logical necessity that if every inertial reference frame measures the same finite maximum speed, they cannot all measure the same time and space (or at least one of the two).

So I’m looking for some sort of mechanism or logical necessity. Could it be relating to something like the equivalence of inertial and gravitational mass? That’s the kind of thing I’m looking for.Of course as you say, the EFEs might be enough if I understand them better.

I would say that the "mechanism" is that the matter, in the form of the stress-energy tensor causes space-time to curve. For instance the Wheeler quote, Spacetime tells matter how to move; matter tells spacetime how to curve.".

I would also say that in this sense, the "mechanism" of both special and general relativity is geometry, by the way.

This of course leads to the question of what makes a geometry "curved". I could say more about this point, but it seems to me that should wait until you're convinced you want to know more about curvature.

Why is it the "stress-energy tensor" that causes space-time to curve and not something else? I would say it is because the stress-energy tensor describes the distribution of matter.

Probably the most direct route to this is to start with the notion that physics can be described by an action principle, which underlies Lagrangian mechanics. Then the variation of the Lagrangian leads to the stress-energy tensor. There are a few wrinkles on the way, which I'd have to refresh my memory on. It's not quite as simple as I make it out to be. But for an overview, I think it's a good start.

There's also an assumption here that one wants to use a tensor formulation. I don't think this is terribly limiting, as I recall anything can be formulated as a tensor. There have been occastions when non-tensor versions of theories have been formulated first, though, with the tensor version being formulated later. It is certainly convenient to learn GR via the tensor formulation as that's the way it's generally taught.

 

[Dale]

if you drop a rock into water, it “curves” the water via displacement (and other things relating to the momentum it imparts during the collision and so on)

If you place a rock in spacetime it curves the spacetime via the EFE (things related to the energy, momentum, and pressure)

 

[PeterDonis]

this is NOT a personal theory, just an example to show what I hope to find: if you drop a rock into water, it “curves” the water via displacement (and other things relating to the momentum it imparts during the collision and so on). I know spacetime isn’t a fluid, so an exact analogy won’t apply, but some sort of cause is what I’m looking for

Then the only answer is "we don't know". Nobody has any model that derives stress-energy curving spacetime from something more fundamental. Perhaps we will have something like that when we have an experimentally confirmed theory of quantum gravity, but we don't have anything like that now.

Certainly as far as GR is concerned, there is no underlying "mechanism": the Einstein Field Equation, which describes how stress-energy curves spacetime, is as fundamental as it gets. (Technically, you could say the EFE is derived from the Einstein-Hilbert Lagrangian, but that Lagrangian doesn't contain any "mechanism" either.)

 

[Dale]

Certainly as far as GR is concerned, there is no underlying "mechanism": the Einstein Field Equation, which describes how stress-energy curves spacetime, is as fundamental as it gets.

I would say it differently. There is a mechanism, that is what the EFE describes.

In his previous example of ripples on a pond, if you want to make that scientific then you would wind up with some set of equations describing how the size/momentum/etc of the rock led to a specific pattern of ripples. A fully described mechanism would consist of a set of governing equations.

In GR that is the EFE

 

[PeterDonis]

There is a mechanism, that is what the EFE describes

You and I accept this as a "mechanism", but apparently the OP does not.

We could just as easily argue that the ripples on the water when you drop a rock in are described perfectly well by a wave equation that does not make use of the microscopic properties of the water molecules, so that equation is the "mechanism" of the ripples. But the OP, in the post I quoted, specifically talks about a "mechanism" that is not just "the wave equation describing the ripples". I interpret that to mean he is looking for something underlying the EFE, and there isn't anything like that in GR. But there might end up being something like it once we have a theory of quantum gravity.

 

[Matterwave]

Well, and this is NOT a personal theory, just an example to show what I hope to find: if you drop a rock into water, it “curves” the water via displacement (and other things relating to the momentum it imparts during the collision and so on). I know spacetime isn’t a fluid, so an exact analogy won’t apply, but some sort of cause is what I’m looking for. Going the other way in GR, that the curvature of spacetime directs the motion of objects along geodesics, does make perfect sense to me, on the other hand.

For what it’s worth, in SR for me an acceptable cause of the relativity of space and time is the invariance of the speed of light, because it’s a logical necessity that if every inertial reference frame measures the same finite maximum speed, they cannot all measure the same time and space (or at least one of the two).

So I’m looking for some sort of mechanism or logical necessity. Could it be relating to something like the equivalence of inertial and gravitational mass? That’s the kind of thing I’m looking for.

Of course as you say, the EFEs might be enough if I understand them better.

If you are looking for a nice derivation of General Relativity from first principles like you can derive special relativity from 1. The principle of relativity and 2. the constancy of the speed of light, you won't find one nearly that clean.

In GR though, you do have motivating principles that "hint at" the EFE's but none through which you can "derive" the EFEs. The principle which most motivates the EFEs is Einstein's principle of equivalence.

Your statement of "equivalence of inertial mass and gravitational mass" is the weak principle of equivalence (it's the one mostly experimentally tested for). The Einstein principle of equivalence goes beyond that statement to say essentially that there is no physical way to determine the difference between a locally free-falling reference frame and a reference frame which is truly free in the vacuum of space away from any gravitational fields (other than the gravity of the laboratory itself - i.e. not accounting for self-gravitating effects). In other words, if I stuck you in a small room that you couldn't see or hear or otherwise send signals out of, you couldn't, even in principle, tell whether you were free-falling toward's Earth at 9.8m/s^2 or whether you were floating out in the middle of space (at least, not until you crashed onto the Earth's surface - when your motion is no longer inertial).

The Einstein principle of equivalence says that not only do all bodies fall at the same rate regardless of their mass, but all physics behaves the same way in these differing reference frames. For example, light, even though it's massless, must still be affected by gravity. If, for example, massless particles are not affected by gravity, then one could differentiate the two differing situations by shining a laser (or a grid of lasers) and seeing if that laser (or a grid of lasers) curved away from you. This equivalence principle therefore suggests that it is in theory possible to describe gravity not as a force that acts on "gravitational mass", but as an inherent curvature in space-time. All material bodies are affected by gravity - not just those that have some "gravitational charge". This is very distinct from the case of e.g. the Electromagnetic force where neutrally charged bodies (that also have no magnetic dipole moment) are not affected by electromagnetism.

Einstein's EFE's which relate (the trace-reverse) Ricci curvature to stress energy directly are a beautiful manifestation of the Einstein equivalence principle, but it is not the unique possible theory that incorporates this principle. One can construct other theories which respect the Einstein principle of equivalence (e.g. Brans Dicke theory), but perhaps Einstein's theory is the most "simple" and elegant of them (this last statement is opinion based).

As a technical aside, there is also a strong principle of equivalence which is even more restrictive than Einstein's (in technical terms, it specifies that the space-time metric is the only field which can affect the space-time curvature). Apparently Einstein's General Relativity is thought to be the only theory which satisfies the strong principle of equivalence - but I am not certain of there being any proof of the fact.

 

[Sorcerer]

Theses are all great answers and I appreciate them all. I don’t want to visualize spacetime as a sort of “ether” or “fabric,” but doing so makes the EFEs make sense as more than just a description. However, until I master the basics of GR I’m going to put off such speculation. For now, the answers given here will have to suffice.

I’m reminded of the recent debate here about what E =mc² actually means. Do you remember? I think it was along the lines of “is mass literally energy, or is this just an equation relating mass to energy via a conversion factor.”

 

[Paul Colby]

In GR that is the EFE

Can't agree more.

One may ask the question on more familiar ground. What is the mechanism that creates electromagnetic fields from charges and currents? The answer is Maxwell's equations. In Maxwell's day mechanical analog models (bedsprings linkages and masses) were proposed for EM. It was all a waste of time.

 

[George Jones]

I have seen the equations relating energy, momentum, pressure/stress to the metric tensor and the curvature of spacetime, as I’m sure most people have. But what is the mechanism?

Suppose Newtonian gravity were the ultimate theory of gravity. What, then, would you say is the "mechanism" that causes gravitational interaction?

In physics in general, once you get to the bedrock of a theory there is no "mechanism", as that implies a lower level of the theory.

 

[LURCH]

This IS a personal theory, but I’ve wondered if the curvature could be the result of some analogue of “surface tension”; the result of extra-dimensional objects (strings, perhaps) punching through our 4-d universe. I bring up this personal theory only to ask if there is a mainstream theory along these lines?

 

[PeterDonis]

if there is a mainstream theory along these lines?

Not that I'm aware of. The "brane" models that occur in string theory have other dimensions with objects ("branes") in them, but AFAIK none of those theories account for the curvature of our 4-D spacetime by invoking such objects.

 

[Sorcerer]

Suppose Newtonian gravity were the ultimate theory of gravity. What, then, would you say is the "mechanism" that causes gravitational interaction?

Oh, Newton’s theory has no mechanism. Even he said that he had no idea. But, I hold out hope for more from GR, since it explains why gravity works (with the EFEs), at least more so than Newton’s model does. I’m just digging for a deeper mechanism, which probably isn’t there.

 

[Dale]

Newton’s theory has no mechanism.

What disqualifies $F=GMm/r^2$ from qualifying as a description of the mechanism? Again, the mechanism for the rock making waves in the water would need to be expressed similarly, so why is one equation a mechanism and the other is not?

 

[Sorcerer]

What disqualifies $F=GMm/r^2$ from qualifying as a description of the mechanism? Again, the mechanism for the rock making waves in the water would need to be expressed similarly, so why is one equation a mechanism and the other is not?

Because the action happens over potentially infinite space with no material or physical mediation instantaneously in that model. Basically my concern is the same one Newton had, that he “left to the consideration” of the reader.Edit- General relativity, as far as I know of it, is obviously a significant upgrade in that respect. But as they say, give an inch and they take a mile.

 

[Paul Colby]

Because the action happens over potentially infinite space with no material or physical mediation instantaneously in that model

The answer to this is the next theory which is currently GR. The answer to your question is the next theory which doesn't yet exist (as far as I can tell).

[edit] Oh, and I wager that this next theory, should it happen in your life time, may well be even less satisfying from a such model points of view.

 

[Dale]

Because the action happens over potentially infinite space with no material or physical mediation instantaneously in that model.

Ok, I can accept that as a reasonable criterion. Basically, you don’t accept instantaneous action at a distance as a mechanism. So the EFE would qualify as a mechanism (or at least would not be disqualified by this criterion) since it doesn’t have instantaneous action at a distance.

 

[Sorcerer]

Ok, I can accept that as a reasonable criterion. Basically, you don’t accept instantaneous action at a distance as a mechanism. So the EFE would qualify as a mechanism (or at least would not be disqualified by this criterion) since it doesn’t have instantaneous action at a distance.

Yeah, it’s a serious upgrade to me. “Energy (etc) warps spacetime by virtue of its presence” makes a lot more sense than “invisible, immaterial magical ray pulls two things that have the same positive ‘charge’ (mass) together as if it were electrical attraction, except instantaneously.”

I suppose one could always ask more “why’s.” But how about this:

Energy density plays a role, right? Momentum density? Stress and pressure?

What does stress, for example, mean in the context of the EFEs? I believe getting a better understanding of what these components of the stress-energy tensor means will help me to see the EFEs as you do- AS the mechanism.

 

[Dale]

What does stress, for example, mean in the context of the EFEs?

Stress is the space-space component of the stress energy tensor. For example it describes the flux of the y component of momentum (a force in the y direction) across a surface normal to x. This is shear stress.

In the EFE this is related to the space-space components of the curvature. The specific type of curvature is a little strange, but basically you can think of the amount that a vector in the x direction changes as you parallel transport it in a plane normal to y.

 

[pervect]

Yeah, it’s a serious upgrade to me. “Energy (etc) warps spacetime by virtue of its presence” makes a lot more sense than “invisible, immaterial magical ray pulls two things that have the same positive ‘charge’ (mass) together as if it were electrical attraction, except instantaneously.”

I suppose one could always ask more “why’s.” But how about this:

Energy density plays a role, right? Momentum density? Stress and pressure?

What does stress, for example, mean in the context of the EFEs?I believe getting a better understanding of what these components of the stress-energy tensor means will help me to see the EFEs as you do- AS the mechanism.

Stress fundamentally means the same thing in physics as it does in mechanical engineering. Though I can't guarantee that engineers talk about it usually exactly the same language as physicists, unfortuately. Hence, for perfect understanding, it's a good idea to look at the math.

For genearl overview purposes, though, if one considers a cantilever beam with a load on the end, the beam is stressed.

Pressure is a form of stress, a form of stress that's the same in all directions (isotropic). If one has a spherical pressure vessel filled with an ideal gas, we'd say that there's stress in the walls of the pressure vessel (specifically, tension), and pressure in the gas. The stress energy tensor is a very general model that can model isotropic stresses (pressures), and non-isotropic ones as well.

It's not particular obvious without going through the math why one needs to include stress in the stress energy tensor, but it wouldn't be a tensor if one didn't include the stress components. Fundamentally, I'd say it's an unexpected consequence of the relativity of simultaneity, so it's not very intuitive.

 

[m4r35n357]

I believe getting a better understanding of what these components of the stress-energy tensor means will help me to see the EFEs as you do- AS the mechanism.

This and this helped me a lot when I first got interested, I think the first one is the closest you will get to a mechanism, maybe you will find it useful.

 

[pervect]

A few more things I can mention. The stress-energy tensor first makes its appearance in special relativity (henceforth SR), so it's needed to understand mechanical problems in SR. A typical SR problem that we've seen on PF that requires the stress-energy tensor would be calculating the angular momentum of an idealized rotating hoop. Be warned, we've also seen a lot of confusion on the part by people not familiar with tensors who try to understand this problem - it's not a particular easy problem to get right without the right methods.

Pedagogically, I would suggest learning about tensors first, before attacking general relativity. It's possible to understand about tensors in isolation, I suppose, but I'd generally recommend a physical application. Applied tensors are usually introduced in electromagnetism. Understanding why the electric field and the magnetic field are combined to form one larger tensor, the electromagnetic field tensor or Faraday tensor, is probably a good first introduction to the use of tensors in physics, and aids in understanding the interrelationships between seemingly different ideas. Understanding this point will aid one in understanding why stress (including pressure) and density are related the way they are in special relativity.

The relationship between stress (including pressure) and density is unexpected, but the principles unifying them into a lager entity are similar to the principles that unify the electric and magnetic fields in electromagnetism. The tensor methods unify the electric and magnetic field into the Faraday tensor - similarly they unify stress, energy, and momentum into the stress-energy tensor.
electromagnetism (EM).

A prerequisite of understanding tensors in general is partial differential equations, henceforth PDE's. PDE's can also be taught in isolation, but are frequently bundled in an applied context, typically Maxwell's equations. The first glimpse one has of Maxwell's equations is presented without tensors. At the graduate level, one introduces tensors and revisits the problem of Maxwell's equations, learning new and powerful techniques.

As a side bonus, one will gain a better understanding of covariance. Tensors transform partial differential equations in a manner that's by definition unaffected by coordinate transformations and/or changes in "frames of reference". I this is a unaswered question for a lot of people, though the mathematical requirements to appreciate how tensors answer this question can be daunting. Tensor methods also play well with Lagrangian methods, as those methods are also about expressing physics in "generalized coordinates". Lagrangian methods are also stressed at the graduate level, though they may be first introduced earlier. It is not essential to learn about Lagrangian mechanics before tackling tensors, however, it can be left until later.