Some questions about general relativity

[faen]

Hi, I have some questions about general relativity. I'd appreciate if somebody could enlighten me :)

I heard that according to Einstein, objects do not accelerate in a gravitational field, it is just spacetime that is warped around them. So what exactly does this mean? Is it that due to relativistic effects, the speed seems to be faster and faster, but instead it is actually time is going slower and slower?

Also if it is true that objects do not accelerate in a gravitational field, how come objects can change speed in the opposite direction. E.g. If i throw a ball up it will change direction and fall down.

Then the next question is, how can space be curved? Isn't it a contradiction to say that space is curved within itself? Or does it exist many layers of space curved relative to each other? Isn't it just easier to say that objects are simply accelerated?

Thanks a lot for any reply :)

 

[nitsuj]

I heard that according to Einstein, objects do not accelerate in a gravitational field, it is just spacetime that is warped around them. So what exactly does this mean? Is it that due to relativistic effects, the speed seems to be faster and faster, but instead it is actually time is going slower and slower?

Also if it is true that objects do not accelerate in a gravitational field, how come objects can change speed in the opposite direction. E.g. If i throw a ball up it will change direction and fall down.

Then the next question is, how can space be curved? Isn't it a contradiction to say that space is curved within itself? Or does it exist many layers of space curved relative to each other? Isn't it just easier to say that objects are simply accelerated?

Thanks a lot for any reply :)

These questions can be answered by reading about geodesics.

 

[faen]

I tried to read it but all I understood is that it just shows a mathematical connection between curved and straight line space? Thanks anyway, but so far I wasn't able to find any answer to my questions.

 

[PeterDonis]

I heard that according to Einstein, objects do not accelerate in a gravitational field, it is just spacetime that is warped around them. So what exactly does this mean?

It means that objects that are moving solely under the influence of "gravity" feel no acceleration: they are weightless, in free fall. Einstein simply adopted that criterion--whether or not an object is weightless--as the *definition* of acceleration. When we need to be precise, we call it "proper acceleration", to distinguish it from "coordinate acceleration"; see further comments below.

However, then we have another phenomenon to explain: tidal gravity. Take two objects dropped from rest at different heights (but with one directly above the other), in vacuum so that there is no air resistance. Both objects are weightless, yet the distance between them increases with time. In special relativity, this is not possible: two objects, both in inertial motion (i.e., weightless), that are at rest at one instant relative to each other, will remain at rest relative to each other forever (so the distance between them will never change). (More generally, objects which are in inertial motion in SR can't change speed relative to each other, so if they are moving at some relative velocity v at one instant, they will move at the same relative velocity v forever.) So special relativity can't be exactly right in the presence of gravity: we need to add something to it.

The something that we add is that spacetime, instead of being flat as it is in SR, is curved, and the curvature shows up as tidal gravity. See further comments on that below.

Also if it is true that objects do not accelerate in a gravitational field, how come objects can change speed in the opposite direction. E.g. If i throw a ball up it will change direction and fall down.

Relative to a coordinate system that is fixed with reference to the Earth, yes, this is true. But the "acceleration" relative to this coordinate system is not proper acceleration; the ball is weightless (assuming we can ignore air resistance), so its proper acceleration is zero. We call the "acceleration" that we see when we adopt coordinates fixed with reference to the Earth (or some other gravitating body) "coordinate acceleration" because it depends on the coordinate system you adopt; if we adopt coordinates that are fixed with reference to the ball, the "acceleration" of the ball disappears.

Then the next question is, how can space be curved? Isn't it a contradiction to say that space is curved within itself?

No, it isn't. First, a key point: what is curved is not space, but *spacetime*. In Special Relativity, spacetime is taken to be flat; the physical meaning of this is, as I said above, that objects which are in inertial motion can't change speed relative to each other. But in the presence of tidal gravity, as we saw above, that isn't the case: two objects that are both in inertial motion *can* change speed relative to each other. So spacetime can't be flat in the presence of gravity: it has to be curved.

It may help in visualizing this to think of a sheet of paper with a coordinate grid on it: one coordinate axis is time, the other is one of the space dimensions--for this discussion we'll assume it's height above the Earth, or more precisely the radial distance r from the Earth's center. Suppose first that the Earth had no gravity: then we could lay our paper with the grid flat, and draw the path of an inertially moving object, like a ball at some height above the Earth, as a straight line on the grid (because with no gravity the ball just floats at a fixed height). Now imagine that we "turn on" the Earth's gravity: General Relativity says that that corresponds to making the sheet of paper curved, in such a way that the ball's path in spacetime bends towards the Earth (its height decreases with time), while still looking like a straight line relative to the grid on the paper. (I emphasize that this is just a simplified version to help with visualization: the actual spacetime curvature due to the Earth is more complicated than this, because spacetime is four-dimensional, not two-dimensional.)

The reason this interpretation works is that all objects "fall" with the same "acceleration" due to gravity (where here "acceleration" means coordinate acceleration). Galileo is supposed to have dropped two cannon balls of different weights from the Leaning Tower of Pisa to show that they would hit the ground at the same time; more recently, one of the Apollo missions dropped a feather and a lump of lead in vacuum on the Moon to show that they would hit the ground at the same time. No other force works this way, and this was one of the chief clues that led Einstein to view gravity as a property of spacetime itself instead of as a property of objects.

Isn't it just easier to say that objects are simply accelerated?

Not if you want to combine the effects of gravity with what we learn from Special Relativity.

 

[nitsuj]

I tried to read it but all I understood is that it just shows a mathematical connection between curved and straight line space? Thanks anyway, but so far I wasn't able to find any answer to my questions.

I hear you faen, I don't have an appreciable understanding of GR either. That is because imo, it strictly requires a very very good understanding of geometries/coordinates/mathematics.

Even the term "curved" is misleading imo, sure it's applicable visually in 1d-2d, but 3 spatial dimensions? And a temporal? It's measurement (geometry) terminology. And imo is misused when applied to our 4D continuum.

I would almost say without a doubt there are other presentations of GR which use terms such as "pressure/displacement" to describe gravity.

GR & curvature are what's popular.

In other words I wouldn't get too caught up in the term "curvature" as a physical description of spacetime, not as a layman.

 

[A.T.]

Thanks anyway, but so far I wasn't able to find any answer to my questions.

Check out these links to get an idea:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html
http://www.relativitet.se/spacetime1.html
http://www.relativitet.se/Webtheses/lic.pdf (Chapter 2)
http://www.adamtoons.de/physics/gravitation.swf

 

[faen]

It means that objects that are moving solely under the influence of "gravity" feel no acceleration: they are weightless, in free fall. Einstein simply adopted that criterion--whether or not an object is weightless--as the *definition* of acceleration. When we need to be precise, we call it "proper acceleration", to distinguish it from "coordinate acceleration"; see further comments below.

However, then we have another phenomenon to explain: tidal gravity. Take two objects dropped from rest at different heights (but with one directly above the other), in vacuum so that there is no air resistance. Both objects are weightless, yet the distance between them increases with time. In special relativity, this is not possible: two objects, both in inertial motion (i.e., weightless), that are at rest at one instant relative to each other, will remain at rest relative to each other forever (so the distance between them will never change). (More generally, objects which are in inertial motion in SR can't change speed relative to each other, so if they are moving at some relative velocity v at one instant, they will move at the same relative velocity v forever.) So special relativity can't be exactly right in the presence of gravity: we need to add something to it.

The something that we add is that spacetime, instead of being flat as it is in SR, is curved, and the curvature shows up as tidal gravity. See further comments on that below.



Relative to a coordinate system that is fixed with reference to the Earth, yes, this is true. But the "acceleration" relative to this coordinate system is not proper acceleration; the ball is weightless (assuming we can ignore air resistance), so its proper acceleration is zero. We call the "acceleration" that we see when we adopt coordinates fixed with reference to the Earth (or some other gravitating body) "coordinate acceleration" because it depends on the coordinate system you adopt; if we adopt coordinates that are fixed with reference to the ball, the "acceleration" of the ball disappears.



No, it isn't. First, a key point: what is curved is not space, but *spacetime*. In Special Relativity, spacetime is taken to be flat; the physical meaning of this is, as I said above, that objects which are in inertial motion can't change speed relative to each other. But in the presence of tidal gravity, as we saw above, that isn't the case: two objects that are both in inertial motion *can* change speed relative to each other. So spacetime can't be flat in the presence of gravity: it has to be curved.

It may help in visualizing this to think of a sheet of paper with a coordinate grid on it: one coordinate axis is time, the other is one of the space dimensions--for this discussion we'll assume it's height above the Earth, or more precisely the radial distance r from the Earth's center. Suppose first that the Earth had no gravity: then we could lay our paper with the grid flat, and draw the path of an inertially moving object, like a ball at some height above the Earth, as a straight line on the grid (because with no gravity the ball just floats at a fixed height). Now imagine that we "turn on" the Earth's gravity: General Relativity says that that corresponds to making the sheet of paper curved, in such a way that the ball's path in spacetime bends towards the Earth (its height decreases with time), while still looking like a straight line relative to the grid on the paper. (I emphasize that this is just a simplified version to help with visualization: the actual spacetime curvature due to the Earth is more complicated than this, because spacetime is four-dimensional, not two-dimensional.)

The reason this interpretation works is that all objects "fall" with the same "acceleration" due to gravity (where here "acceleration" means coordinate acceleration). Galileo is supposed to have dropped two cannon balls of different weights from the Leaning Tower of Pisa to show that they would hit the ground at the same time; more recently, one of the Apollo missions dropped a feather and a lump of lead in vacuum on the Moon to show that they would hit the ground at the same time. No other force works this way, and this was one of the chief clues that led Einstein to view gravity as a property of spacetime itself instead of as a property of objects.



Not if you want to combine the effects of gravity with what we learn from Special Relativity.

Thanks, that helped me progress in my understanding. I find it strange that Einstein made the criterion of how weightlessness isn't proper acceleration. Personally I've always imagined gravity as that it simply accelerates each particle equally. The feeling of weight comes from our particles sharing their energy between each other, but in the case of simultaneous acceleration in the same direction they won't share this energy among themselves, and thus not have any effect on our nerve or contain any information about the total collection of particles.

I'm still trying to understand how exactly curved spacetime can create motion. No matter how slow time is going, how can it make an object turn and move backwards?

 

[faen]

I hear you faen, I don't have an appreciable understanding of GR either. That is because imo, it strictly requires a very very good understanding of geometries/coordinates/mathematics.

Even the term "curved" is misleading imo, sure it's applicable visually in 1d-2d, but 3 spatial dimensions? And a temporal? It's measurement (geometry) terminology. And imo is misused when applied to our 4D continuum.

I would almost say without a doubt there are other presentations of GR which use terms such as "pressure/displacement" to describe gravity.

GR & curvature are what's popular.

In other words I wouldn't get too caught up in the term "curvature" as a physical description of spacetime, not as a layman.

Hm yeah you're right.. The term pressure sounds more logical to me too. I'm still wondering how it's possible.. I think I finally understood special relativity, but this is even harder :)

 

[faen]

Check out these links to get an idea:
http://www.physics.ucla.edu/demoweb..._and_general_relativity/curved_spacetime.html
http://www.relativitet.se/spacetime1.html
http://www.relativitet.se/Webtheses/lic.pdf (Chapter 2)
http://www.adamtoons.de/physics/gravitation.swf

Thanks, studying it now :)

 

[nitsuj]

I think I finally understood special relativity, but this is even harder :)

Yea the man himself gave the comparable SR=child's play.

I'd be surprised if his "SR" paper took more than a year & on his own.

GR 10 yrs & with much collaboration (specifically math, without doubt the concept was much more easily had...at least for him)

 

[Nugatory]

I'm still trying to understand how exactly curved spacetime can create motion. No matter how slow time is going, how can it make an object turn and move backwards?

Here's an analogy: You're holding an apple. There's an ant walking in a circle around the "equator" of the apple. As you watch the ant, you'll see him first moving left to right, than as he follows the curvature of the skin of the apple he'll be moving away from you until he disappears around the back of the apple... and then comes back into sight again moving towards you... Even though the ant believes himself to be walking in a straight line on a two-dimensional apple skin, he ends up reversing direction because the skin is curved.

This is an analogy because it's working with curved two-dimensional space instead of curved four-dimensional space-time. But at least we can visualize it, and we can't visualize our paths through even a flat four-dimensional space time, let alone a curved one.

 

[harrylin]

Hi, I have some questions about general relativity. I'd appreciate if somebody could enlighten me :)

I heard that according to Einstein, objects do not accelerate in a gravitational field, it is just spacetime that is warped around them. [..]

Instead of listening to such hear-say, a good starting point may be his "popular" book:
https://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory

In that book he also gives some examples of common warped coordinate spaces.

The main criticism that I have against it, is that he is sometimes a bit fuzzy - some things are even less clearly explained than in his "non-popular" paper (which contains a lot of complex math, but the text around the math is very readable):
https://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity

PS this one may also be helpful (and it is, as it promises, rather brief!):
https://en.wikisource.org/wiki/A_Brief_Outline_of_the_Development_of_the_Theory_of_Relativity

 

[A.T.]

http://www.relativitet.se/Webtheses/lic.pdf (Chapter 2)

Thanks, studying it now :)

This is more extensive than the link above, also chapter 2:
http://www.relativitet.se/Webtheses/tes.pdf

 

[harrylin]

Thanks, that helped me progress in my understanding. I find it strange that Einstein made the criterion of how weightlessness isn't proper acceleration. [..]

I have not seen that criterion by Einstein - Peter, reference please!

As a matter of fact, I am still searching for a paper in which Einstein discussed the physical interpretation of free fall in a real (non-uniform) gravitational field.

 

[pervect]

I'm still trying to understand how exactly curved spacetime can create motion. No matter how slow time is going, how can it make an object turn and move backwards?

I don't understand why you think an object is turning around and "moving backwards", and I've got an uneasy feeling when I hear you say "how slow time is going". Both of these things have little to do with space-time curvature.

Let's start with some basics. We can represent space-time with a space-time graph. And space-time curvature can be (loosely) interpreted as saying "we need to draw our space-time diagrams on a curved surface, rather than a flat one".

So, let's take a curved surface, a sphere, and ddraw a space-time diagram on it, and see what happens. We'll say that on this space-time diagram, "north" represents the time direction, just as we usually draw time going "up" on the flat space-time diagram.

So, suppose we have two people at the equator, 1 nautical mile apart, and they both go north What happen?

http://upload.wikimedia.org/wikipedia/commons/a/a2/Geodesic_deviation_on_a_sphere.svg

Well, what happen is that they get closer and closer together. Not only that, but they appear to accelerate towards one another - initially, the relative rate of change of distance between them, their relative velocity, is zero, but as time goes on they approach each other more and more rapidly, just as if they were accelerating towards one another.

It's a bit like they were gravitationally attracting each other - or , perhaps another analogy, being attracted by the tidal gravity of some external source.

But neither of them is doing anything more than moving along a straight line on a space-time diagram.

Anyway, the generic name for what I've just described is "geodesic deviation". Geodesics are just the straightest possible lines on a curved surface - if you read about GR, you'll probalby be hearing a lot more aobut them.As far as books go, I' suggest something along the lines of "Exploring black holes" I've heard mixed reviews about Schutz, "Gravity from the ground up", but it's another possibility as it's pretty low level from what reviews I did read. I suspect you'll find the modern treatments more illuminating than Einstein's original work, but - it's hard to predict for sure. If you've got the time, try reading various sources until you find one that you understand.

"Exploring black holes" has a few introductory chapters online, some of which cover things like curvature.

http://www.eftaylor.com/download.html#general_relativity

Ben Crowell also has an online book, "Light and Matter". I've only read sections of it. To avoid the appeareance of advertising, I'll let you find "Light an Matter" with a web search, if you're interested.

 

[harrylin]

[..] space-time curvature can be (loosely) interpreted as saying "we need to draw our space-time diagrams on a curved surface, rather than a flat one".

So, let's take a curved surface, a sphere, and ddraw a space-time diagram on it, and see what happens. We'll say that on this space-time diagram, "north" represents the time direction, just as we usually draw time going "up" on the flat space-time diagram.

So, suppose we have two people at the equator, 1 nautical mile apart, and they both go north What happen?

http://upload.wikimedia.org/wikipedia/commons/a/a2/Geodesic_deviation_on_a_sphere.svg

Well, what happen is that they get closer and closer together. Not only that, but they appear to accelerate towards one another - initially, the relative rate of change of distance between them, their relative velocity, is zero, but as time goes on they approach each other more and more rapidly, just as if they were accelerating towards one another.[..].

That's one of the clearest illustrations that I have seen - so much better than the rubber sheet which misses the main issue.

 

[Warp]

I heard that according to Einstein, objects do not accelerate in a gravitational field, it is just spacetime that is warped around them.

I'm a complete layman when it comes to physics, and it's very hard for me to fully visualize this in my head as well, but this is the best way to describe it that I have come up with. (Note that it's most probably a gross simplification and quite inaccurate, but it should give you a grasp of how it works.)

In order to understand how this works, we need to visualize 4-dimensional space. This is, of course, extremely hard for the human brain to grasp, so what we do is to compress one of the dimensions to zero length (in other words, flattening the 3-dimensional space into a plane) and replace the now "unused" third dimension with the original fourth time dimension.

So you have to imagine the entire three-dimensional space compressed into a plane. Therefore the Earth is a disc, and the ball is a (much) smaller disc slightly apart from it. Movement in space is now restricted to this plane.

What is happening is that this plane (which represents space) is moving along the time axis (which in this visualization would be the third axis, perpendicular to the plane.) So everything moves with the plane: The Earth, the ball, everything.

If mass had no effect on the geometry of spacetime, then nothing would change. The ball would simply be where it is and the Earth would be where it is, as both traverse the time axis. However, masses bend spacetime, which means that the time axis does not consist of straight lines, but curved lines. (In reality this is much more complex because the spatial dimensions are also bent, but for the sake of simplicity let's forget about that here.)

Because the Earth bends the time axis, it causes the ball to approach the Earth as both move in this time axis. (Basically, the ball is simply moving, due to inertia, along the shortest path that exists in the time axis.)

If the ball was "stationary" to begin with, it will not move on the spatial plane at first, but the curvature of the time axis will cause it to make a parabolic path towards the Earth (in other words, making it move in the spatial plane directly towards the center of the Earth disc.)

If instead of starting stationary you were to throw the ball parallel to the surface of the Earth, then in this scenario it would have an initial movement on the space plane (parallel to the edge of the Earth disc). As it moves in the time axis, it will follow the curve that bends towards the Earth as well as retaining its horizontal motion, thus following a different path in this three-dimensional visualization.

(The only question that I do not really grasp yet is why everything moves in the time axis.)

 

[PeterDonis]

I find it strange that Einstein made the criterion of how weightlessness isn't proper acceleration. Personally I've always imagined gravity as that it simply accelerates each particle equally.

But this "accelerates each particle equally" is frame-dependent; if you are freely falling yourself, you won't see other freely falling objects "accelerate"; they will just float next to you. We are so familiar now with video from the Space Shuttle and the International Space Station, that shows the astronauts and objects around them all floating weightless, that it may be hard to imagine how much of an insight it was to realize in 1907, as Einstein did, that "if a person falls freely, he will not feel his own weight". (Einstein later called this "the happiest thought of my life".)

The fact that we can make the "acceleration due to gravity" disappear by falling freely ourselves is why Einstein wanted a different definition of acceleration, one that wouldn't be dependent on adopting a particular point of view. We see freely falling objects "accelerate" because we're at rest on the surface of the Earth; but our theory of physics shouldn't be dependent on being at rest on the surface of the Earth. Defining acceleration as proper acceleration--i.e., as feeling weight--is general: it works everywhere, and doesn't depend on adopting a particular point of view.

 

[PeterDonis]

(The only question that I do not really grasp yet is why everything moves in the time axis.)

Because "moving in the time axis" is just "existing"; it's just moving from now towards tomorrow.

 

[Warp]

No matter how slow time is going, how can it make an object turn and move backwards?

If you think about the visualization I described in my previous post, throwing a ball upwards would be (in that visualization) having an initial speed of the ball disc away from the Earth disc. As both discs move in the time axis (which is being curved by the Earth), the ball will follow a parabolic path due to its own inertia (because that's the shortest path in this curved coordinate system.) In our three-dimensional "slice" (which would be the plane in the visualization) it looks to us like the ball just goes up (away from the Earth) and then comes down, when in reality it's following a parabolic spacetime curve.

Because "moving in the time axis" is just "existing"; it's just moving from now towards tomorrow.

That doesn't really explain anything... :/

 

[PeterDonis]

That doesn't really explain anything... :/

What would count as an explanation? Do you think you can somehow avoid "moving in time" from now to tomorrow? That's what the "time axis" represents: one point on that axis is your "now", another one is your "tomorrow", and since you can't avoid moving from now to tomorrow, you can't avoid moving along the time axis from one point to another.

 

[Warp]

What would count as an explanation?

Something that helps me understand, or even get a faint idea.

Do you think you can somehow avoid "moving in time" from now to tomorrow?

There's nothing that forces something to move in the three spatial axes, so why is the time axis different, even though for the calculation of movement due to gravity it's basically handled as just a fourth geometry axis?

The three spatial axes are free to be traversed in any direction at any speed, but the fourth time axis is not. You cannot travel it backwards, you cannot stop. "Something" forces you to travel it in one direction, and your speed is determined by something else than your acceleration (basically mass determines it, rather than anything else, although I may be really wrong here.)

So it's both handled as "just another dimensional axis", but it's also quite different from the other three. This is, AFAIK, a fundamental aspect of GR, but I just cannot grasp it.

That's what the "time axis" represents: one point on that axis is your "now", another one is your "tomorrow", and since you can't avoid moving from now to tomorrow, you can't avoid moving along the time axis from one point to another.

I know, but I don't understand why. What is it that makes everything traverse the time axis?

 

[PeterDonis]

There's nothing that forces something to move in the three spatial axes, so why is the time axis different, even though for the calculation of movement due to gravity it's basically handled as just a fourth geometry axis?

Because you can't stay at the same point of time the way you can stay at the same point of space. That's just a physical fact. If you want to know why that physical fact is a fact, well, nobody knows. So perhaps there is no "explanation" that will meet your requirements in this particular case.

The three spatial axes are free to be traversed in any direction at any speed, but the fourth time axis is not. You cannot travel it backwards, you cannot stop. "Something" forces you to travel it in one direction, and your speed is determined by something else than your acceleration (basically mass determines it, rather than anything else, although I may be really wrong here.)

Actually every object's "speed through time" is the same (technical point: this is true for every object with nonzero rest mass; things like light that have zero rest mass work a bit differently). Mass, or more precisely energy, is like "momentum through time", not "speed through time"; your "speed through time" is your "momentum through time" (i.e., your energy), divided by your mass (i.e., your energy), so it ends up being the same for everything.

I know, but I don't understand why. What is it that makes everything traverse the time axis?

There may not be any answer to this beyond "that's just the physical fact". See above.

 

[Warp]

Because you can't stay at the same point of time the way you can stay at the same point of space. That's just a physical fact. If you want to know why that physical fact is a fact, well, nobody knows. So perhaps there is no "explanation" that will meet your requirements in this particular case.

I'm not sure I'm satisfied with that answer because it just sounds like "yeah, in principle you could travel in the time axis at will, but for some reason that nobody knows, you are forced to travel in one direction and you can't travel the other way" even though, if I have understood correctly, that's not what GR postulates. AFAIK being able to "travel" freely in the time axis would cause all kinds of paradoxes and would break GR. I think that GR handles time differently from the spatial axes (but I know nothing about the GR equations, so it all goes well above my head.)

 

[PeterDonis]

I'm not sure I'm satisfied with that answer because it just sounds like "yeah, in principle you could travel in the time axis at will, but for some reason that nobody knows, you are forced to travel in one direction and you can't travel the other way" even though, if I have understood correctly, that's not what GR postulates.

GR doesn't really postulate anything as far as whether or not you can travel along the time axis at will like you can along the spatial axes. It models spacetime as a 4-dimensional thing that just "is", not as something "moving" through the time axis. Interpreting the time dimension as something that objects "move" through is a way of linking up the spacetime model with our everyday experience, but you can do all the math and make all the predictions in GR without ever using it.

AFAIK being able to "travel" freely in the time axis would cause all kinds of paradoxes and would break GR. I think that GR handles time differently from the spatial axes (but I know nothing about the GR equations, so it all goes well above my head.)

Mathematically, the reason the time axis is different is that it has an opposite sign in the metric. This is true in SR as well; take a look at the Minkowski line element:

d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2

The dt^2 term has opposite sign from the other terms; that means the interval d\tau^2 is not positive definite. That is, the interval between two distinct points can be positive, zero, or negative. That's not possible in ordinary Euclidean geometry: there, the distance between two points can only be zero if the points are identical, and it can never be negative.

What all this means is that, in spacetime, there is something fundamentally different about a timelike interval with d\tau^2 > 0, vs. a spacelike interval with d\tau^2 < 0 or a null interval with d\tau^2 = 0. They are three physically distinct kinds of intervals. The same is true in GR; the only difference there is that the line element can look different than the formula above, due to spacetime curvature.

But you'll notice that nowhere in any of this did I talk about anything "moving" along a curve, or through an interval. If two points are separated by a timelike interval, that means some timelike curve connects them, so some object's worldline can pass through both points. But that's just a fact about the geometry of spacetime, in the same way that the statement "the Earth's equator passes through Quito, Ecuador and Nairobi, Kenya" is a fact about the geometry of the Earth's surface. (I don't know that that's exactly a fact, btw; those cities are close to the equator but probably not exactly on it. But it illustrates what I'm getting at.) We can describe the geometry of spacetime without talking about anything "moving" in it, just as we can describe the geometry of the Earth without talking about any objects moving on it.