Initial Impetus Needed for Curved SpaceTime?

[abrogard]

In a gravitational field you can put a particle and it is immediately subject to a force that tries to accelerate it.

But in a curved spacetime notion if you put a particle it has no reason to move. Right?

 

[houlahound]

It would move to lower it's energy?

 

[Drakkith]

But in a curved spacetime notion if you put a particle it has no reason to move. Right?

Sure it does. It is already moving through spacetime and the curvature will tend to cause the spatial distance between the particle and the source of the curvature to decrease.

 

[abrogard]

why is it already moving?

 

[Drakkith]

To be clear, it's moving through the time portion of spacetime. As for why, I haven't the slightest idea. I don't know if it's just an intrinsic property of objects and spacetime or if there's an underlying reason.

 

[abrogard]

i guess that's unavoidable, that it moves through the time portion. just existing predicates that I suppose. so I should have specified my question is about why would it move through the spatial portion of spacetime?

 

[Drakkith]

Imagine you and I start walking side by side in a direction perpendicular from the equator towards the north pole. As we walk, we notice that although neither of us are turning or moving sideways relative to the Earth's surface, we are somehow getting closer and closer together. The reason is that the surface of the Earth is curved, not flat. This analogy takes place on the 2d surface of a 3d object, but the concept applies to spacetime as well. The geometry of spacetime, which is what GR describes, causes the particle to move.

 

[abrogard]

yes, imagine we start walking. now imagine we don't start walking but just stand there.

 

[secur]

The "walking' is analogous to moving through time. You can't stand still, but must move through time at "1 second per second" - as we all know from experience.

I don't know if it's just an intrinsic property of objects and spacetime or if there's an underlying reason.

- No distinction. There may be an "underlying reason" why it's an intrinsic property of objects in spacetime, but it's still an intrinsic property. Admittedly that's more a philosophical point than physical.

 

[Drakkith]

yes, imagine we start walking. now imagine we don't start walking but just stand there.

But spacetime is not the surface of the Earth. You don't have an option of not moving through spacetime. You simply do, and there's nothing you can do to stop it. I don't think there's any underlying reason why. I think it's just an observation and a critical foundation of GR.

 

[abrogard]

they draw spacetime diagrams which allow for movement through time without movement through space. sure there's always movement through spacetime but the x co-ordinate can remain fixed while the y moves. surely?

 

[Drakkith]

they draw spacetime diagrams which allow for movement through time without movement through space. sure there's always movement through spacetime but the x co-ordinate can remain fixed while the y moves. surely?

I think so. If you're in space you can use a rocket engine to avoid moving in a spatial direction, relative to other objects, even in curved spacetime. The upward force from the ground prevents you from falling towards the Earth, so you aren't moving relative to it even under gravity. But the key here is that there has to be a force that prevents this. Under free-fall conditions, where the only thing acting on you is gravity, you cannot avoid moving through space.*

*Technically I think "motion" should be replaced with some kind of "acceleration", but I don't know GR well enough to elaborate.

 

[secur]

Let's back up a bit and consider Special Relativity first - flat Minkowski spacetime.

You're always moving at a fixed rate "1 second per second" (i.e., "time passes"). When motionless in space, your "motion vector" projects entirely into the time dimension. But when you experience acceleration and therefore move in space, from the point of view of an external observer, the vector tilts into the 3 space dimensions ("Lorentz boost"). However it maintains its invariant length, so you don't travel as far in time. That's one way of understanding time dilation: when moving, your "local time" slows down.

From your own POV you're always in your own "rest frame" and never move in space, only time, at the fixed rate "1 second per second". What seems to be motion is actually the rest of the universe moving relative to you. Thus your vector doesn't "tilt"; instead, the rest of universe tilts around you, causing motion through space.

In GR it gets complicated by curved spacetime. Without a force acting you still don't move (in space) - same as SR. But gravity is not a force, rather it's the curvature itself; and you're moving along that curvature, in your own time direction (on a geodesic). Another object, also experiencing no force, no longer must remain stationary relative to you, as with SR, because the curvature at its location might be different. If your two vectors are converging, the result is, you move towards that object. Drakkith's analogy of walking on the Earth demonstrates this. In free-fall you still aren't moving in your local space dimensions; but the curvature causes you to approach the gravitating object because your time direction is pointed (normally, just a little) towards the gravitating object.

By leaving out details this explanation may not be quite right technically but I think it's pretty close.

 

[abrogard]

you say you still don't move and then go on to say you are moving.

Without a force acting you still don't move (in space) - same as SR. But gravity is not a force, rather it's the curvature itself; and you're moving along that curvature,

That sentence is difficult for me to understand.

Without x no z.
But y is not x.
y is q
you z on q

what?

 

[Nugatory]

they draw spacetime diagrams which allow for movement through time without movement through space. sure there's always movement through spacetime but the x co-ordinate can remain fixed while the y moves. surely?

Yes. Just choose your coordinate system so that the x-axis is perpendicular to the path through spacetime of the object and the y-axis is not, and the y coordinate will change while the x coordinate does not.

 

[A.T.]

i guess that's unavoidable, that it moves through the time portion. just existing predicates that I suppose. so I should have specified my question is about why would it move through the spatial portion of spacetime?

This might help:

 

[secur]

Without a force acting you still don't move (in space) - same as SR. But gravity is not a force, rather it's the curvature itself; and you're moving along that curvature, in your own time direction

That sentence is difficult for me to understand.

Without x no z.
But y is not x.
y is q
you z on q

Without x no z.

- x = "force"
- z = "move in SPACE"

But y is not x.

- y = "gravity"

y is q

- q = "curvature"

you z on q

- z = "move in TIME" - your two z's are not the same, need different labels. call this one T

The only confusion is, how does movement in time wind up moving in space? That's exactly the question Drakkith's analogy answers. Go through it again.

You and me are at the equator 69.2 miles apart - one degree of longitude. Then we both travel due North, staying strictly on our own longitude.

North, along the longitude, is analogous to the time direction. Latitudinal movement "sideways" is analogous to movement in our local space coordinates. So in this analogy, we move only in TIME (T) not SPACE (z)

When we get halfway up the Earth, latitude 45 degrees, we'll only be about 35 miles apart; and we'll meet at the North Pole where our lines converge - can't avoid it if we keep traveling North. So we wound up moving sideways, towards each other, but not "on purpose".

That's analogous to falling into a black hole singularity. Suppose we start at the event horizon 96.2 miles apart; halfway, we'll be 35 miles apart (making various assumptions); and we'll meet at the singularity. Neither of us moved in local space coordinates but our forward time directions inevitably converged.

 

[m4r35n357]

To the OP

You and me are at the equator 69.2 miles apart - one degree of longitude. Then we both travel due North, staying strictly on our own longitude.

North, along the longitude, is analogous to the time direction. Latitudinal movement "sideways" is analogous to movement in our local space coordinates. So in this analogy, we move only in TIME (T) not SPACE (z)

When we get halfway up the Earth, latitude 45 degrees, we'll only be about 35 miles apart; and we'll meet at the North Pole where our lines converge - can't avoid it if we keep traveling North. So we wound up moving sideways, towards each other, but not "on purpose".

That's analogous to falling into a black hole singularity. Suppose we start at the event horizon 96.2 miles apart; halfway, we'll be 35 miles apart (making various assumptions); and we'll meet at the singularity. Neither of us moved in local space coordinates but our forward time directions inevitably converged.

To take it one step further, imagine that instead of physically walking north, we are each standing still on a moving walkway traveling at the same speed northwards. We see each other approach over time as if by magic. We call that "gravity".

 

[abrogard]

I'll think it over. It all seems very much like trickery.

This curvature you talk about is a special case. Not all curves are spheres.

Things are apparently happening just as in the case of conventional gravity - particles falling towards a mass. From all around. 360 degrees. As they near the mass they will obviously get closer together. The gravity is pulling them to that mass. That's where the gravity is. The lateral coming together is incidental. If there's lateral gravitational effect because of their masses that would distort those paths and that closing.

Your black hole is the same as that.

You seem to be saying the gravity is the lateral closing. The whole thing about black holes is the gravity is in it.

You essentially then are saying you cannot move in time without moving in space. A special movement. A gross movement. Not the movement of electrons, not brownian motion, but a gross movement in the direction towards the nearest mass.

It is very puzzling. Can't you put it simpler than that?

Have you people a very clear understanding of the essential nature of reality in this respect?

I ask because it appears to me some essential features of reality cannot be clearly understood. They are clear because the maths makes them clear. Or put it another way: the maths is clear. But then to have a clear intellectual understanding of what this represents is quite a different thing.

Example down at my low level is the Lorentz contraction. I can follow the high school math argument put by Ramamurti Shankar in his youtube lectures. And I can see the contraction come out. I'm convinced by the maths. But there's no way I can intellectually understand what is really going on. I can't see why it is contracting. Though I could even measure it.

Can you clearly see this notion of warped spacetime?

 

[houlahound]

Mental pictures are irrelevant to nature.

You are arrogant to think nature is constructed just so you personally can obtain a mental picture of it.

Do the hard math if you want understanding.

 

[abrogard]

I am simply trying to ask if it is possible to have an understanding beyond a mathematical one. Or is contemplating the maths as good as we get? That was my intent with the anecdote about the Lorentz contraction.

[mentor's note: unnecessary digression removed to keep the thread on topic]

 

[Nugatory]

I am simply trying to ask if it is possible to have an understanding beyond a mathematical one. Or is contemplating the maths as good as we get?

For gravity and general relativity, it's actually fairly easy to develop an intuitive understanding. You have to train yourself to think in terms of objects following paths through four-dimensional spacetime instead of moving freely in three-dimensional space and to shift between the two viewpoints. The first step is to practice some with the two-dimensional spacetime diagrams... What does the spacetime diagram of two objects approaching one another look like?

 

[secur]

abrogard, I'm glad you don't accept it just because everybody else does; some physicists don't really get it either.

There's a difference between understanding it, and believing it. I learned this stuff as mathematics: differential geometry. The professor told us, vaguely, that it had something to do with GR, but nobody cared. We did Gaussian curvature in 3 dimensions; Riemann's 4-d version; Levi-Civita, etc - as mathematics. The only physicist mentioned was Weyl - not Einstein. As math it stands alone, you don't have to "believe" anything.

But when you learn it as physics you have to "believe" gravity works this way or else it's all nonsense. If I'd approached it as physics I would have flunked the course, because I didn't believe it back then.

You say, this is not really different from Newton's picture of gravity. The BH attracts objects by the force of gravity, they fall into it, why do we need "curved spacetime"? Well, here are the key facts that can't be explained that way.

First: It's experimentally proven that your "local time" slows down at high speeds, and in a gravitational well. Newton's approach can't deal with that - everyone's time must be the same (absolute time). Without this vital linkage between space and time, they would never have come up with relativity or Minkowski 4-d space.

Second: The way the universe is expanding. At great distances, all the galaxies are apparently moving away at very high speeds. Curved, expanding spacetime gives a sensible, satisfactory explanation. Otherwise, it's incomprehensible that all those galaxies are actually running away from us at near c! Admittedly other explanations are possible but this is a key factor convincing me that GR is real.

Finally, GR also explains gravitational waves. Try to fit those facts into the Newtonian picture! It can be done but requires incredible contortions.

On the other hand: singularities are merely a convenient fiction until the real explanation comes along - don't try to believe in those. Also there's a question whether GR (and, Newtonian) gravity is correct for large distances - even ten billion miles, much less a million light years. In fact all the observational evidence is against it. Dark Matter was invented to explain those discrepancies, but they still haven't detected any. But these details don't detract from the basic idea of curved spacetime.

As for the analogy: you're right; as far as I've described it, it can be explained by ordinary Newtonian gravity, because I didn't go into the time distortions. After all there's nothing mysterious about the walkers on the longitude lines coming together, either. The thing that's hard to get - and, to get across - is that the forward motion in space is analogous to the sideways (latitudinal) motion on Earth, NOT walking North. Walking North is analogous to getting older (through time), NOT falling towards the singularity. But there's no use beating it to death.

Can you clearly see this notion of warped spacetime?

It's impossible to visualize 4 dimensions, we can only visualize 3. Some famous people claimed to have that ability (Lewis Carroll, Salvador Dali, others), but I don't believe them. So, sure I can see it clearly, but only in 2 or 3 dimensions. For instance, 2-d spacetime diagrams are simple enough, putting aside the issue whether they're "true". It's just differential geometry, what's the big deal?

My advice, study the math as math. Read Gauss and Riemann's original work to see pure genius in action. Keep thinking about the evidence that I find convincing. If you're like me, after a while, you'll begin to suspect that maybe Einstein knew what he was talking about.