3 dimensions of space and 1 of time

[atyy]

I do understand the difference between a wave and the medium through which the wave travels. However, I'm not sure that gravity and spacetime can really be separated like that.

Yes, generally you cannot split spacetime up into background + wave. Misner, Thorne and Wheeler say it is like a wave on the ocean, which is the wave and which is the ocean? But sometimes splitting up the ocean into wave and ocean is quite a good approximation. Similarly, such a split is sometimes quite a good approximation of the full mathematics of general relativity.

So, if gravity is described as geodesics, then light/matter, whatever, must have momentum in order to follow those geodesics. right?

You have the right idea - if you talk about 4-momentum, not 3-momentum. The curvature is that of spacetime, and we are always moving in spacetime, because we are always moving in time.

 

[Hoku]

Further down the rabbit hole, things get curiouser and curiouser. Let me piece together things I've picked up:

1) Frame Dragger said, "Light follows a geodesic INTO the black hole... It is not "dragged" in..."

2) A.T. said, "[an apple] still advances in space-time, and that is where it follows a geodesic..."

3) atyy said, "...we are always moving in spacetime, because we are always moving in time."

4) New Scientist says, "...whenever anything...tries to travel through the universe... it follows a trajectory that is curved... The result of this...is what we think of as gravity.

5) In this thread, https://www.physicsforums.com/showthread.php?p=2617230#post2617230, Maurol2 says, "What is needed is a dynamical theory of gravity."

What I'm piecing together here, is that gravity has no more power than a hill. A hill is a noun, not a force. It seems like momentum is the only real force in play and things with momentum - in space or in time - follow geodesics because it's the easiest, quickest path. Could this be why they can't make gravity fit with the other 3-forces? I'm no longer certain that gravity IS a force.

 

[A.T.]

It seems like momentum is the only real force in play and things with momentum - in space or in time

You have a default "movement" in space-time, which cannot be stopped, but only changed in direction:
http://www.adamtoons.de/physics/relativity.swf

follow geodesics because it's the easiest, quickest path.

Follow geodesics, means go straight ahead like a car without steering. It is only locally the "easiest, quickest" path.


In the http://www.relativitet.se/spacetime1.html" a toy car is used to model geodesics:

 

[Frame Dragger]

I love it when these things come together! Hoku, I do believe you've gotten it! Thois is fun!

 

[atyy]

What I'm piecing together here, is that gravity has no more power than a hill. A hill is a noun, not a force. It seems like momentum is the only real force in play and things with momentum - in space or in time - follow geodesics because it's the easiest, quickest path. Could this be why they can't make gravity fit with the other 3-forces? I'm no longer certain that gravity IS a force.

Test particle GR:
T1 Energy-momentum curves spacetime
T2 Test particles move on spacetime geodesics

Test particle GR is a very good approximation, but there is a flaw in it. What is a test particle? It is something which has no energy-momentum and does not curve spacetime. So all real particles cannot obey the test particle equation T2, they should obey T1.

Also, most matter is not point particles, at least not in GR. Point particles are black holes in GR. So in GR, everything is a field, which matches up with our view of all the other forces and matter - such as the electromagnetic field (a gauge force field) and the electron field (a matter field), and it is in fact possible to make a quantum theory of gravity.

The problem with quantum general relativity is that it doesn't seem hold at very, very high energies (this is not proven yet, it may turn out that gravity is asymptotically safe). But neither does quantum electrodynamics. So the geometric test particle view of GR is not the reason for the failure of quantum general relativity at very, very high energies.

 

[Hoku]

You have a default "movement" in space-time, which cannot be stopped, but only changed in direction:

The real mystery with this is the "time" part. I did look at the "free fall" page you gave a link to a few times last night. I even kept it open and looked again this morning. The problem that I'm having with it is that it let's space be infinite but time ends at, what, .11s, .12s? Obviously, time cannot be less infinite than space.

In this graph, it seems the only reason the event returns to 0.0m is because of this limitation imposed on time, which closes the curved surface. The adamtoons link you included in your last post seems better because it doesn't make time finite. BUT, then it doesn't become clear why the object should return to 0.0m.

Test particle GR:
T1 Energy-momentum curves spacetime
T2 Test particles move on spacetime geodesics

Test particle GR is a very good approximation, but there is a flaw in it. What is a test particle? It is something which has no energy-momentum and does not curve spacetime. So all real particles cannot obey the test particle equation T2, they should obey T1.

This is confusing for me because, if a test particle has no energy-momentum, then why should it obey T1 any better than T2, aside from proving that it doesn't curve spacetime? Or is that the point?

Point particles are black holes in GR. So in GR, everything is a field...it is in fact possible to make a quantum theory of gravity.

I can see the connection between gravity and general relativity, but you seem to be using these interchangably, which is confusing me. From what I understand, gravity is a component of GR, but GR is ultimately describing relationships between different moving things. Gravity is just a path to follow, right? On it's own, gravity has little to do with relationships. Even when something is following gravity there is still no "relativistic value" without the the movement of something else to compare it with. Do you see my confusion? So, to help clear this up, I'd ask if we're trying to make QM fit gravity or relativity.

 

[Gatchaman]

The real mystery with this is the "time" part. I did look at the "free fall" page you gave a link to a few times last night. I even kept it open and looked again this morning. The problem that I'm having with it is that it let's space be infinite but time ends at, what, .11s, .12s? Obviously, time cannot be less infinite than space.

Spacetime does end. That's why its called a singularity.

 

[Gatchaman]

From what I understand, gravity is a component of GR, but GR is ultimately describing relationships between different moving things. Gravity is just a path to follow, right? On it's own, gravity has little to do with relationships. Even when something is following gravity there is still no "relativistic value" without the the movement of something else to compare it with. Do you see my confusion? So, to help clear this up, I'd ask if we're trying to make QM fit gravity or relativity.

I think the problem you have is that you're thinking too much of Newtonian gravity and somehow this still exists as a force in general relativity (GR). In GR, gravity is a purely geometrical/stress-energy construction. There are no "forces" involved.

 

[atyy]

This is confusing for me because, if a test particle has no energy-momentum, then why should it obey T1 any better than T2, aside from proving that it doesn't curve spacetime? Or is that the point?

I should have said - a test particle is a particle whose energy momentum does not curve spacetime. There are no such things. In real life, we treat a real particle as a test particle if its energy-momentum is so small compared to the rest of the universe that its contribution to spacetime curvature is negligible.

I can see the connection between gravity and general relativity, but you seem to be using these interchangably, which is confusing me. From what I understand, gravity is a component of GR, but GR is ultimately describing relationships between different moving things. Gravity is just a path to follow, right? On it's own, gravity has little to do with relationships. Even when something is following gravity there is still no "relativistic value" without the the movement of something else to compare it with. Do you see my confusion? So, to help clear this up, I'd ask if we're trying to make QM fit gravity or relativity.

Yes, I would say that general relativity is not a theory of relativity (but this can be debated). General relativity is a theory of gravity in which the spacetime metric is affected by mass-energy and is the gravitational field. In comparison, in special relativity, the spacetime metric is not affected by mass-energy.

 

[A.T.]

from : http://www.relativitet.se

The real mystery with this is the "time" part. I did look at the "free fall" page you gave a link to a few times last night. I even kept it open and looked again this morning. The problem that I'm having with it is that it let's space be infinite but time ends at, what, .11s, .12s?

No, time doesn't end there, it continues on a new layer of the diagram. Think of the diagram as a roll. You can extend this diagram along both dimensions into infinity.

In this graph, it seems the only reason the event returns to 0.0m is because of this limitation imposed on time, which closes the curved surface.

No, the limitation of the displayed time interval has nothing to do with the shape of the world lines. They are just affected by the local curvature.

The adamtoons link you included in your last post seems better because it doesn't make time finite. BUT, then it doesn't become clear why the object should return to 0.0m.

Here is the more complex version:
http://www.adamtoons.de/physics/gravitation.swf
Keep in mind that this diagram is also multilayer like a roll.

It shows the space-time along an axis all the way trough to the other side of a massive sphere, and the world line of a test particle moving on this axis. You can set:
initial position : 1
initial speed : 0.16
To get a body thrown up vertically and come down again.

Press "Help" for more info.

 

[Hoku]

Spacetime does end. That's why its called a singularity.

Whether or not spacetime actually "ends" is in the realm of philosophy. Your statement is presumptious. Fortunately, whether or not spacetime actually has an end is irrelevant to my point for 2 reasons:

1) It hasn't ended YET so we cannot close the curve.
2) Since spacetime is a "singularity", as you point out, then space and time both need to be closed on the graph simultaneously to maintain consistency. Space cannot extend beyond time.

No, the limitation of the displayed time interval has nothing to do with the shape of the world lines.

I think it does. The world line is defined by geodesics, right? Closing the curved surface affect it's geometry, which, in turn affects geodesics.

This is bringing me back to the question of why we actually stick to the Earth. Why do objects return to 0.0m? I did click the help button on the new adamtoons site that A.T. last posted. In the help menu, under the "samples" heading, it says "When you push play, you see in the 3-D view, how the object is moving along its world line. The world line changes it's direction in regard to the dimensions because it is taking the straightest possible way." But this makes me ask:

1) The straightest possible way to where?
When it is answered, "Back to the Earth, of course." Then I need to ask:
2) Why does it want to get back to the Earth?
From this, I expect the answer to be, "because it's the straightest possible way."

Do you see what I'm having trouble with? Circular reasoning.

A.T. said movement in space-time cannot be stopped but it can change direction. Evidentally, that's not entirely true. As I said early in this thread, we only really have control over 2-dimensions. We need planes, rockets, etc., to gain control of the third. Why? Why are we stuck to the Earth?

I might pose this question in a new thread in hopes of finding others that haven't been following this one to help answer. But I will let it sit here for a minute to see how far we can get it.

 

[Gatchaman]

Whether or not spacetime actually "ends" is in the realm of philosophy. Your statement is presumptious. Fortunately, whether or not spacetime actually has an end is irrelevant to my point for 2 reasons:

1) It hasn't ended YET so we cannot close the curve.
2) Since spacetime is a "singularity", as you point out, then space and time both need to be closed on the graph simultaneously to maintain consistency. Space cannot extend beyond time.

No philosophy here. Just your lack of understanding GR in a mathematical manner.

 

[Mentz114]

we only really have control over 2-dimensions. We need planes, rockets, etc., to gain control of the third.

I can walk up a hill without rocket assistance.

Why? Why are we stuck to the Earth?

Oh, the futility of human existence !

Ah, you mean literally. Well, there's this thing called gravity you see, blah blah ...

 

[Gatchaman]

I'm no longer responding to this thread. I really think Hoku is trolling now. All of the mumbo-jumbo metaphysics he throws in here along with legitimate GR contributions from other forum members just doesn't sit right with me.

Good luck.

 

[Gatchaman]

Hoku,

As easy as this may sound for me to say:

Go grab an introductory book about General Relativity and start learning the MATH. Without knowledge of the MATH, you will have absolutely no understanding of GR.

But, I think you need to grab yourself an introductory physics book to review basic Newtonian Laws. Its obvious from your statements that you lack the basic knowledge.

 

[Frame Dragger]

Way to hold to post #64!

 

[Hoku]

I can walk up a hill without rocket assistance.

I'm going to assume that you're serious with this statement and respond accordingly. I think a 2-dimentional surface is allowed to have "hills". I also think that a Flatlander can "ascend" those hills. Does this mean Flatlanders have control over the 3rd dimension? I don't think so.

"Why? Why are we stuck to the Earth?"

Oh, the futility of human existence !

Ah, you mean literally. Well, there's this thing called gravity you see, blah blah ...

Thanks for the comic relief, Mentz! That should carry me through the day...

Gatchaman, this is a second request for you to withdraw from this thread. I will report you for harrassment if you continue.

 

[JesseM]

Whether or not spacetime actually "ends" is in the realm of philosophy. Your statement is presumptious.

No, it's just a statement of how general relativity deals with cases like black holes. GR may well be wrong about this, but aren't we just discussing the theory's predictions on this thread, not speculating about what might be true in a theory of quantum gravity that supercedes it?

1) It hasn't ended YET so we cannot close the curve.
2) Since spacetime is a "singularity"

Spacetime is not a singularity. In GR spacetime can contain singularities at certain points like the center of black holes and the Big Bang, and according to GR these are the only points where worldlines can "end".

I think it does. The world line is defined by geodesics, right? Closing the curved surface affect it's geometry, which, in turn affects geodesics.

A.T. was just giving a diagram which is an imperfect representation of actual 4D spacetime geometry. The fact that time "ends" at a certain point on the surface in the diagram doesn't reflect how it actually works in GR, although the diagram may capture other aspects of it.

This is bringing me back to the question of why we actually stick to the Earth. Why do objects return to 0.0m?

If our bodies were solely affected by gravity (meaning we all followed geodesic paths at all times) we would fall right through the surface of the Earth and pass near the center at high speed and then pop up on the opposite side of the Earth, following something like a highly eccentric orbit around the center. But other forces can prevent objects from following geodesics, in this case the electromagnetic force between the ground and our feet which prevents us from falling down towards the center.

I did click the help button on the new adamtoons site that A.T. last posted. In the help menu, under the "samples" heading, it says "When you push play, you see in the 3-D view, how the object is moving along its world line. The world line changes it's direction in regard to the dimensions because it is taking the straightest possible way." But this makes me ask:

1) The straightest possible way to where?
When it is answered, "Back to the Earth, of course."

No, the straightest possible path on the curved spacetime around the Earth, where the spacetime curvature is determined by the distribution of mass and energy in this region of spacetime according to the Einstein field equations, mainly by the mass of the Earth itself.

A.T. said movement in space-time cannot be stopped but it can change direction. Evidentally, that's not entirely true. As I said early in this thread, we only really have control over 2-dimensions. We need planes, rockets, etc., to gain control of the third. Why? Why are we stuck to the Earth?

Here you are talking about mere questions of biology, not physics. If we didn't have legs (if we were plants for example), then we wouldn't have "control" over 2 dimensions. If we had wings (if we were birds or pterodactyls for example) we would have "control" over 3 (and obviously we already have a limited sort of control since we can jump). This sort of control is always a matter of electromagnetic interactions between parts of our body (legs, wings) and the environment (ground, air) which allow us to move along non-geodesic paths.

 

[Mentz114]

Thanks for the comic relief, Mentz!

You started it. Some of the stuff you're saying is ludicrous. Don't take my post as agreement. I can move around in 3 dimensions. I suspect you'll start pushing some crackpot theory soon.

 

[Dale]

"When you push play, you see in the 3-D view, how the object is moving along its world line. The world line changes it's direction in regard to the dimensions because it is taking the straightest possible way." But this makes me ask:

1) The straightest possible way to where?
When it is answered, "Back to the Earth, of course." Then I need to ask:
2) Why does it want to get back to the Earth?
From this, I expect the answer to be, "because it's the straightest possible way."

Do you see what I'm having trouble with? Circular reasoning.

It is not circular reasoning, but it does require a bit of understanding what it means for a line to be "straight". There are essentially two different but equivalent ways of determining if a path is straight. The first (global) definition is that a straight path is a path which locally minimizes the distance between the start and the end (the shortest distance between two points is a straight line). To use this one you need a beginning point and an ending point. The second (local) definition is that a straight path is a path which doesn't turn anywhere (parallel transports its tangent vector). To use this one you need a beginning point and an initial tangent direction.

So, how do we understand this idea wrt gravity? First, we consider the path of a particle to be represented by some line in spacetime, a particle at rest will be represented by a line in the time direction, while a particle which is moving will be represented by a diagonal line with a slope which is related to the velocity. Inertially moving particles travel along geodesics. So, if we hold our hand still and drop a rock, the event of our dropping the rock is a point in spacetime and at that point the rock is at rest so its worldline's tangent is in the time direction. From that point, it travels straight (in the local sense), parallel transporting its own tangent vector through spacetime, until it hits the ground, at which point it is no longer traveling inertially.

Gatchaman, this is a second request for you to withdraw from this thread. I will report you for harrassment if you continue.

Good luck with that. I looked back and couldn't find a single thing that Gatchaman said that was out of line.

 

[Frame Dragger]

I don't think Hoku is a crackpot... I think he's been EXPOSED to them, or he's had limited information about GR, and what he didn't know he's tried to fill in by himself. Silly, but then, that's why some people come here to learn.

I'm hardly the sweetest dear in these things, but while he's not getting it, I don't believe it's intentional. I think we may be dealing with a language barrier, and a Kurger-Dunning Effect. Remember how that KDE can be ADDRESSED by gradually teaching the person about the subject in question.

@Hoku: Here's your chance to prove me right, or very wrong: Before we started discussing them here, were you familiar with the terms "Worldline" "Tangent" "Interia" "Geodesic" "time-like" "space-like" and "Scalar/Vector/Tensor"? If not... if this is you trying to fit in, while learning: STOP. You have a lot to UNlearn, and then a lot to learn. You're at a point (trust me here) where good people like JesseM, DaleSpam, Mentz and others I've seen help other people, are giving up on you, or thinking you're a kook.

If you have an alternate theory, present it... if not, and you're just lost in this (but maybe you don't realize how much), please... start asking BASIC question ABOUT the basics you need to learn FIRST. You're either trying to run when you should be learning to walk, or you're screwing around with all of us. I BELIEVE you're trying to run, but I think I'm in the minority now.

I'm really not used to being the last person to have a scrap of hope for someone! :sigh:

 

[atyy]

"Interia"

Is that why we're stuck to the earth?

 

[Frame Dragger]

Is that why we're stuck to the earth?

I thought that was sticky tape and layers of turtles! AHHH my worldview! :Rofl:

 

[Hoku]

DaleSpam, thanks for isolating the relevant issue and approaching it respectfully (I'm especially grateful to atty, also).The issues we're discussing are interesting but don't seem to be touched on in the laymans literature of relativity that I've come across. Everywhere I look it says that relativity has to do with how spaceships look to each other as they're passing. But in this discussion it turns out that relativity is just about describing the geometry of spacetime. Everywhere I look it says that gravity is one of 4 forces. But in this discussion it turnes out that gravity isn't a force at all. These misconceptions are what I came into this thread with and what people have helped me change. However, these new perspectives bring lots of questions that were never an issue when gravity was a force and relativity was spaceships. As a result, I've been searching online to understand all this better but somehow my questions keep being missed and/or the language is too complicated. Example, "geodesics are defined to be curves whose tangent vectors remain parallel if they are transported along it." This might be fine for you, but it's a little complicated for a layman. How about this one, "A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow." "Huh?" Geodesics are not complicated. "An ant walks around an orange. When you cut its path from the peel, you find it is a straight line. This is a geodesic" "Oh, I see!" That's why I keep coming back to this thead asking the questions I do and why I'm grateful for those who have maintained some "compassion" for my journey.

The first (global) definition is that a straight path is a path which locally minimizes the distance between the start and the end (the shortest distance between two points is a straight line). To use this one you need a beginning point and an ending point. The second (local) definition is that a straight path is a path which doesn't turn anywhere (parallel transports its tangent vector).

Am I right in thinking that the "global definition" describes a null geodesic and the "local definition" describes the timelike worldlines you mentioned in post #32?

...a particle at rest will be represented by a line in the time direction...[a dropped rock's] worldline's tangent is in the time direction.

This seems to be supporting what atty had said about matter pressing into the Earth because it maintains momentum in the time direction. I'm fairly certain that the next sentence will get everyones eyes rolling because it probably means I haven't been paying proper attention. What confuses me about this is why our "time momentum" insists we go only one direction in space - "earthwards". Isn't time momentum just momentum into the future? I guess I just don't understand why the future can't include the "upwards" direction. Maybe there is a book that you can recommend that describes these different geodesic/worldlines. I think the null geodesic is easier to visualize than the "timelike worldline".

Good luck with that. I looked back and couldn't find a single thing that Gatchaman said that was out of line.

Post #35 was unnecessary and just plain mean - especially when he already knew I was feeling down. What value does that serve aside from taunting like a bully in the schoolyard?

In light of post #35, post #40 seems no better. Did he even read what I was presenting? What value is there to post #40 aside from having made a snotty remark? I had asked that he abandon the thread shortly after this because I'm putting forth a strong and sincere effort here. I don't need to wade through his unproductive negativity.

Post #64 was unnecessary. I already asked him to leave. Why did he need to "excuse" himself aside from using it as a reason to accuse me of "trolling" and spewing "metaphyiscal mumbo-jumbo"? These are insults with no value beyond that of harrassment.

Frame Dragger may be mean, but tied in with it is an invitation to step up to the plate. He also includes helpful ideas. That's why I'll put up with him. Gatchaman, on the other hand, has offered nothing of value and all I feel from him is harrassed.

 

[Hoku]

Thanks FD.

By the way, I know it shouldn't matter but somehow I keep being bothered by being called a "he".

 

[Frame Dragger]

Thanks FD.

By the way, I know it shouldn't matter but somehow I keep being bothered by being called a "he".

Oh! Sorry, I didn't realize, she it is.

 

[atyy]

I think gravity, as conceived in general relativity, is a force - but that can wait till later. For the moment, let's take gravity to be not a force, it is the curvature of spacetime and free-falling particles (ie. those subject to gravity alone, and not to electromagnetism) follow geodesics of spacetime.

It is important to remember that the geodesics are of spacetime, not "space". And yes, they can include what we call the "up" direction of "space". Take a look at the figures on pages 66 and 67 of http://people.maths.ox.ac.uk/~nwoodh/gr/gr03.pdf . It's a bit too advanced (especially since you have to flip back quite a few pages to figure out what p and u mean), but just a pointer for the moment.

 

[JesseM]

As a result, I've been searching online to understand all this better but somehow my questions keep being missed and/or the language is too complicated. Example, "geodesics are defined to be curves whose tangent vectors remain parallel if they are transported along it." This might be fine for you, but it's a little complicated for a layman. How about this one, "A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow." "Huh?" Geodesics are not complicated. "An ant walks around an orange. When you cut its path from the peel, you find it is a straight line. This is a geodesic" "Oh, I see!" That's why I keep coming back to this thead asking the questions I do and why I'm grateful for those who have maintained some "compassion" for my journey.

OK, when DaleSpam said "A geodesic is a locally length-minimizing curve", here are some simple examples to show what that means. You know how in a flat 2D plane, the shortest distance between any two points is a straight line, right? Well, all geodesics are straight lines in a 2D plane. Likewise, on the surface of a sphere, if you pick two points on the surface what would be the shortest path between them? Turns out that if you find the great circle that has both points on it (a great circle is a circle whose center is the center of the sphere, like the equator or a line of longitude on a globe), then the section of the great circle that lies between the points is the shortest possible path on the sphere between them. So, that's the geodesic path on a sphere.

On bumpy surfaces with more complicated curvature, we can distinguish between globally geodesic paths and locally geodesic ones. A globally geodesic path is the shortest distance between two points, period. A locally geodesic path is one where, if you look at any "nearby" paths that stay close to the main path but deviate slightly from it, they will always be a bit longer than the geodesic one.

In when you're dealing with paths through spacetime rather than space, there's the added twist that a geodesic path is actually the one that measures the greatest amount of time (locally at least, when compared to other 'nearby' paths), not the shortest amount of distance. This has to do with a mathematical difference in how you calculate time along paths through spacetime, where time is not treated quite the same way as a spatial dimension. But you don't have to worry to much about this, there's still a close analogy between the idea of geodesics in spacetime and geodesics on spatial surfaces.

On the other hand, when DaleSpam said "geodesics are defined to be curves whose tangent vectors remain parallel if they are transported along it", just think about the ant walking on the curved surface again--if you know the ant's starting point, and you know which direction it starts walking in (which is basically the ant's 'tangent vector' at that point), then assuming the ant is doing its best to walk in a straight line, that's enough info to know what its geodesic path will look like extending away from that point. So similarly, if you know something's position at a particular moment and its initial speed and direction, that'll tell you what its geodesic path in curved spacetime will be.

 

[Dale]

Everywhere I look it says that relativity has to do with how spaceships look to each other as they're passing. But in this discussion it turns out that relativity is just about describing the geometry of spacetime.

That is mostly an historical artifact. When Einstein first developed relativity he did a lot of "thought experiments" (an oxymoron) comparing things from the point of view of different trains. Shortly after that Minkowski took the theory and cast it into a unified geometric framework that is considered to be "modern relativity" by people who actually use it. However, for some strange reason "pop-sci" treatments of relativity invariably attempt to "modernize" relativity by changing from trains to rockets but otherwise sticking to the "thought experiment" approach that is more than a century out of date by now.

Example, "geodesics are defined to be curves whose tangent vectors remain parallel if they are transported along it." This might be fine for you, but it's a little complicated for a layman.

This is all just geometry, and because you are familiar with the curved geometry on a sphere (just talking about the 2D surface, no 3rd dimension) most of this is understandable. For parallel transport just think of someone who walks around always trying to keep their arm pointed in the same direction at each step. If he starts on the equator with his arm pointing east and walks due north along the prime meridian, then when he gets to the north pole his arm will be pointing south (obviously) along the 90 deg east line. Instead, if he starts in the same location but starts by heading east (arm straight in front) for 90 deg and then turns north (arm to the right) when he reaches the north pole his arm will point south along the 180 deg line. Even though both cases kept their arm pointed in the same direction at each step they wound up pointed in different directions due to the curved geometry of the sphere. This is the idea of parallel transport.

So a geodesic is a path which parallel transports it tangent vector. That means that you start at some point, stick your arm out in front of you and walk that direction always keeping your arm pointed the same direction at each step and always walking that direction at each step.

Am I right in thinking that the "global definition" describes a null geodesic and the "local definition" describes the timelike worldlines you mentioned in post #32?

No, both the local and global definitions apply to any kind of geodesic. They are equivalent definitions.

What confuses me about this is why our "time momentum" insists we go only one direction in space - "earthwards". Isn't time momentum just momentum into the future?

That is due to the direction of the curvature around the Earth (Schwarzschild metric). Spefically, time runs slower at your feet than at your head (see AT's diagram where the lower circles are wider than the higher circles). This curvature in the time direction geometrically means that timelike geodesics curve downwards.

Think again about the geometry on the surface of a sphere. We will think of the north-south direction as representing time and the east-west direction as representing space. Two nearby objects which are at rest at the equator both heading in the time direction (north) will intersect at the north pole despite each line being everywhere straight and following a geodesic (longitude lines are great circles). Because of the curvature they will always "attract" each other.

 

[Hoku]

Wow! JesseM just gave a heaping mound of mashed potatoes, atty thwapped on a pound of stuffing and DaleSpam topped it off with ladles of gravy. That's quite a bit to work on. Thanks for these helpful posts. I don't want this post to get too long so I'm just addressing JesseM in it. Afterwards, I'll work on another post to address atty and Dale.

A locally geodesic path is one where, if you look at any "nearby" paths that stay close to the main path but deviate slightly from it, they will always be a bit longer than the geodesic one.

I don't see how this is different from the global path. If a global path is "the shortest distance, period", then of course any nearby path you look at will be longer than it.

Now, DaleSpam's post says that they are "equivalent definitions". So, what is the value of differentating them?

In when you're dealing with paths through spacetime rather than space, there's the added twist that a geodesic path is actually the one that measures the greatest amount of time (locally at least, when compared to other 'nearby' paths), not the shortest amount of distance.

Did you mean to say "shortest" amount of time (as opposed to greatest)? This reminds me of driving my car during rush hour. The freeway is definitely the shortest distance to get from A to B, however, during rush hour, taking the streets can still get you from A to B in the shortest time. Is this along the lines of what what you're saying?

This has to do with a mathematical difference in how you calculate time along paths through spacetime, where time is not treated quite the same way as a spatial dimension. But you don't have to worry to much about this, there's still a close analogy between the idea of geodesics in spacetime and geodesics on spatial surfaces.

This actually seems like a critical piece of information to me. You say I don't have to worry too much about it, and you may be right for the moment, however, when I think of spacetime as a uniform thing without obstructions, I have trouble imagining how the fastest way in time could be different from the fastest way in space.

I can make this idea fit with my "driving" analogy, though. Having to go around buildings and follow different one-way streets etc, etc., impose limits on where you can travel in space to get to B... Hmm... I think might have made a good connection here:

We are timelike because we're trying to get to our "destination" via the fastest "time" route. The problem is, the Earth (building) is in the way so we are stuck on it. If we were not "timelike" we could "choose" to use a "spacelike" route and go AROUND the Earth. But we're not, so we keep trying the futile route. Is this right?

 

[Frame Dragger]

Wow! JesseM just gave a heaping mound of mashed potatoes, atty thwapped on a pound of stuffing and DaleSpam topped it off with ladles of gravy. That's quite a bit to work on. Thanks for these helpful posts. I don't want this post to get too long so I'm just addressing JesseM in it. Afterwards, I'll work on another post to address atty and Dale.

Wow... I want turkey now. I think I'm actually going to order a turkery dinner... and I was just telling someone else about suggestability. Mmmm... mashed potatoes...

 

[Dale]

I don't see how this is different from the global path. If a global path is "the shortest distance, period", then of course any nearby path you look at will be longer than it.

Do you understand the difference between a global minimum* and a local minimum of a function?
http://en.wikipedia.org/wiki/Maxima_and_minima

A geodesic is a local minimum, but there may be more than one geodesic connecting two events, and one of those may be shorter than the other. For example, consider two points on a sphere, the shortest distance between those two points is along a great circle, but you can go either way around the great circle. Each great-circle path is a geodesic, but generally one will be shorter than the other. However, even the longer great circle path is a local minimum meaning that if you almost follow the long great circle but deviate just a little then your distance will increase.

*I have been using the word "minimum" but if I were being rigorous I should use the word "extremum" to reflect the fact that a geodesic can be a minimum or a maximum or their higher-dimensional counterparts.

 

[Hoku]

Before I begin this, I want to clarify a couple of things. First, when I was giving examples of complicated language at the top of post #75, I wasn't referring to DaleSpam or anyone on this thread. I should have made that more clear. I was generally trying to show why my Google searches ("internet") can lead to dead ends for me and why you all are important. Second, I'm GOING to run into technical language that is above my head. While I encourage you to go easy on this language if you can, as long as there is otherwise some grain of understandability to what you're saying, I'm happy to Google the language and figure it out. That's what I've been doing. atty's response #78 really challenges my ability to figure out this language, and I think that's great! It shows that he has some faith in me and it helps me prove to you that I really am putting in the work. In the end, though, I know my efforts on this one will be more comedy for you. That's ok, I'm laughing about it myself. Hopefully this won't turn out to be a real major road block.

I think gravity, as conceived in general relativity, is a force...

The plot thickens...Now, some people might slap their foreheads to atty's remark thinking, "atyy, you're just confusing the poor girl!" To that, I would have to say... YOU'RE RIGHT! However, because atty adds, "For the moment, let's take gravity to be not a force", I am required to relinquish both my confusion and judgements on the matter and proceed with an open mind. Still, I'm grateful for this comment because it holds the promise of something neat to be learned ahead.

Take a look at the figures on pages 66 and 67 of http://people.maths.ox.ac.uk/~nwoodh/gr/gr03.pdf . It's a bit too advanced (especially since you have to flip back quite a few pages to figure out what p and u mean), but just a pointer for the moment.

Ok, I opened the link and found the figures. You're right, it's pretty advanced. So, I'm scrolling up through the pages trying to pick out "P's" and "u's". There's a-lot of them to be found! Did I find the right "u" in the middle of page 29? It says that it is a parameter of a surface in space. That seems pretty obvious just by looking at the figure. But then the top of page 47 it says "since the speed of u is so small...", which confuses me because I didn't think a surface could have speed. On the bottom of page 35 it say "u" is a function on spacetime, which seems to be in agreement with the definition on page 29.

I'm laughing right now because this is really like asking me to find a needle in a haystack and I'm sitting here picking through pieces of straw. I slowly scrolled all the way up to page 15 scanning each page for "u's" and "p's". Considering that every page might as well be Chinese to me, my handicap in this is increased.

I got stuck trying to find the right "p" because, at first, I was looking for lower cases. To that end, I found, at the bottom of page 15, a definition that makes it a "rest density". But, then I realized I needed a capital p. So I scanned up again looking for capital p's this time. At the top of page 17 is says P is a particle. But this can't be the right p. Actually, when I read about the figure on page 62, I think I must've been right about it being lower case. It's not very clear on figures on page 66 or 67.

On a more positive note, this document does have some english in it, which makes it a limited resource for me. I'll print out some of the pages and see if they help. I really liked the lightcone diagram on page 19. It doesn't immediately help, but the image will stay in my mind and may come in handy later on.

 

[atyy]

Before I begin this, I want to clarify a couple of things. First, when I was giving examples of complicated language at the top of post #75, I wasn't referring to DaleSpam or anyone on this thread. I should have made that more clear. I was generally trying to show why my Google searches ("internet") can lead to dead ends for me and why you all are important. Second, I'm GOING to run into technical language that is above my head. While I encourage you to go easy on this language if you can, as long as there is otherwise some grain of understandability to what you're saying, I'm happy to Google the language and figure it out. That's what I've been doing. atty's response #78 really challenges my ability to figure out this language, and I think that's great! It shows that he has some faith in me and it helps me prove to you that I really am putting in the work. In the end, though, I know my efforts on this one will be more comedy for you. That's ok, I'm laughing about it myself. Hopefully this won't turn out to be a real major road block.


The plot thickens...Now, some people might slap their foreheads to atty's remark thinking, "atyy, you're just confusing the poor girl!" To that, I would have to say... YOU'RE RIGHT! However, because atty adds, "For the moment, let's take gravity to be not a force", I am required to relinquish both my confusion and judgements on the matter and proceed with an open mind. Still, I'm grateful for this comment because it holds the promise of something neat to be learned ahead.

Ok, I opened the link and found the figures. You're right, it's pretty advanced. So, I'm scrolling up through the pages trying to pick out "P's" and "u's". There's a-lot of them to be found! Did I find the right "u" in the middle of page 29? It says that it is a parameter of a surface in space. That seems pretty obvious just by looking at the figure. But then the top of page 47 it says "since the speed of u is so small...", which confuses me because I didn't think a surface could have speed. On the bottom of page 35 it say "u" is a function on spacetime, which seems to be in agreement with the definition on page 29.

I'm laughing right now because this is really like asking me to find a needle in a haystack and I'm sitting here picking through pieces of straw. I slowly scrolled all the way up to page 15 scanning each page for "u's" and "p's". Considering that every page might as well be Chinese to me, my handicap in this is increased.

I got stuck trying to find the right "p" because, at first, I was looking for lower cases. To that end, I found, at the bottom of page 15, a definition that makes it a "rest density". But, then I realized I needed a capital p. So I scanned up again looking for capital p's this time. At the top of page 17 is says P is a particle. But this can't be the right p. Actually, when I read about the figure on page 62, I think I must've been right about it being lower case. It's not very clear on figures on page 66 or 67.

On a more positive note, this document does have some english in it, which makes it a limited resource for me. I'll print out some of the pages and see if they help. I really liked the lightcone diagram on page 19. It doesn't immediately help, but the image will stay in my mind and may come in handy later on.

p and u are defined on p60. u is essentially 1/radius, so any orbit that intersects u=0 is either coming in from infinitely far away, or escaping to infinitely far away.

Beware rho, which looks like p

 

[Hoku]

You're not going to learn GR this way.
But, its completely entertaining and laughable!

Obviously, the diagram atty suggested was unsuccessful. What I demonstrated was how much I'm willing to put into this. I can imagine you thinking, "If you're willing to put so much into it, go back to school." I'd LOVE to! I enjoy learning. Unfortunately, we all can't have everything and sacrifices are unavoidable in a life. Consequently, I've been restricted to self study. And I'm not the only one. Layman science books have a pretty good market. Unfortunately, there are some important gaps in what they present.

The crux of the problem being addressed in this thread is whether or not gravity is a force. Is it or isn't it (rhetorical)? If it isn't, then how does it work... and why? In my Google research, I've come across this question from other laymen who are as confused as I. They don't seem to be making much more progress than I, either. This problem is the root of many annoying assumptions like, "a black hole moves faster than light". I've read sites that try to address this confusion but they ultimately fail in clearing it up. They give answers that may content some, but if you keep thinking about it, that content isn't enough.

My "usefulness" here is that I'm not a quiter and maintain my focus in the face of such opposition as I've received here. Lots of other people don't care enough to fight through it, so the "ignorance" that annoys you so much remains. You may think my questions and conclusions are stupid, but they are natural lines of reasoning for someone at my level. I have 2-years of college, with more chemistry credits than any other dicipline - I loved chemistry (but not the lab). I also have a varied personal library. I began pre-calculus but had to quit school mid-quarter. That was, what, 15-years ago?

I believe it is possible for a layman to understand this issue. Math is applicable to the real world. That's its usefulness! So we really need to find real world symbolism to help resolve the problem. There are lots of layman that want it and there are lots of annoyed "science advisors" that would like it, too. Many of you are "properly" educated. That means you sat through classes and read books. But it ALSO means you had teachers, tutors and classmates to discuss these things with and ask questions to. Laymen don't have those resources, but many of us still have the passion to know.

I'd imagine it's easy to isolate yourself in a click of math-gabbers, but there is value to expanding your realm of symbolism beyond math language. Reaching out to laymen is a good way to do this. You might be surprised at the new neural connections you feel happening in your own brain as a result.

I expect to be curb-smashed for this post so I will end it here.

 

[atyy]

To make yourself confused, why don't you try lecture 5 by Bertschinger http://ocw.mit.edu/OcwWeb/Physics/8-224Exploring-Black-Holes--General-Relativity---AstrophysicsSpring2003/LectureNotes/index.htm ? He starts off with "How would you describe general relativity to your parents?" ... "Gravity is spacetime curvature, whatever that means" ... "Many physicists have the impression, the mistaken impression that ... gravity is no longer a force".

More accessible on Youtube:
http://www.youtube.com/watch?v=8MWNs7Wfk84&feature=PlayList&p=858478F1EC364A2C&index=2

 

[Dale]

The crux of the problem being addressed in this thread is whether or not gravity is a force.

I disagree on this, I don't think this is a key question here. Whether or not "X is a Y" always depends strongly on the definition of Y. In relativity a lot of Newtonian terms are re-defined so that they have different meanings. So saying that gravity is or is not a force does not help your learning until you understand what is meant by the word "force" in this contex. You have much larger conceptual hurdles to overcome.

 

[dx]

The crux of the problem being addressed in this thread is whether or not gravity is a force.

See post #8 in this thread: https://www.physicsforums.com/showthread.php?t=382066

 

[Hoku]

In relativity a lot of Newtonian terms are re-defined so that they have different meanings. So saying that gravity is or is not a force does not help your learning until you understand what is meant by the word "force" in this contex.

In response to this, I've tried finding "re-definitions" of the word "force" as it applies to relativity. The search has come up fruitless. All definitions of a force as it relates to gravity are equivalent to every other definiton with the exception that gravity, like electromagnetism, and the strong and weak forces, is not a "contact force".

DaleSpam, you're suggesting that gravity can still be defined as a force but many people disagree with this, including this government website. http://www.Newton.dep.anl.gov/askasci/ast99/ast99099.htm , a Dept. of Energy outreach for k-12 educators and their students.

See post #8 in this thread: https://www.physicsforums.com/showthread.php?t=382066

dx, you and Nabeshin seem to be taking similar, diplomatic approaches to the problem by trying to overlook semantics. I think this is a noble approach, but it also seems like a cop-out. In post #10, Nabeshin says, "Gravity does not even exist in GR. So it makes no sense to speak of it as a force." Then he says, "No mention of the word gravity is ever needed." In other words, let's not mention the ugly stepchild and pretend that everything's ok.

http://www.uoregon.edu/~struct/courseware/461/461_lectures/461_lecture4/461_lecture4.html says:
"A "force" is an action that changes, or tends to change, the state of motion of the body upon which it acts." In other words, a force does work, right? So, I'm wondering if the relevant question might be, "what has energy?"

Why are planets in orbit? Isn't it because of opposing forces? Isn't it similar to tying a ball at the end of a string and spinning it in a circle? Energy, thus force, is required to resist something, isn't it?

Why don't planets take a geodesic path right into the sun? Isn't it because they have their own agenda - their own energy - that is trying to go somewhere else but the gravity prevents them from leaving?

 

[Hoku]

To make yourself confused, why don't you try lecture 5 by Bertschinger http://ocw.mit.edu/OcwWeb/Physics/8-224Exploring-Black-Holes--General-Relativity---AstrophysicsSpring2003/LectureNotes/index.htm

As atty already included in his post, professor Bertschinger is acknowleging that people are not seeing gravity as a force. Bertschinger thinks this perception is a mistake. After going through different equations he concludes that "In the weak field limit, Einstein field equations = Newtonian equations for gravitational potential." Does this help resolve anything?

 

[Altabeh]

In response to this, I've tried finding "re-definitions" of the word "force" as it applies to relativity. The search has come up fruitless. All definitions of a force as it relates to gravity are equivalent to every other definiton with the exception that gravity, like electromagnetism, and the strong and weak forces, is not a "contact force".

Gravity is an interacting force between matter and the fabric of spacetime but in GR it is not actually described in such a way that a curious reader is able to dig the straight meaning of "force" from the context at first glance. Simply gravity is treated like a "geometrical object" that is only observed through the changes in the shape of spacetime which we call it "curvature". In the Newtonian physics, this has a different facet to look through which is like you're now feeling something is pushing you down to the ground so the "gravitational force" exists and of course this has something to do with the fact that Newtonian theory is not a geometrized framework to work in but rather is a classical theory dealing with the ordinary implications of time, space, force and etc.

dx, you and Nabeshin seem to be taking similar, diplomatic approaches to the problem by trying to overlook semantics. I think this is a noble approach, but it also seems like a cop-out. In post #10, Nabeshin says, "Gravity does not even exist in GR. So it makes no sense to speak of it as a force." Then he says, "No mention of the word gravity is ever needed." In other words, let's not mention the ugly stepchild and pretend that everything's ok.

I've already encountered with Nabeshin's argument before though I'm a little bit uncomfortable with it! I think we can't say gravity does not exist in GR because then we have no curvature and nothing to talk about! Also in the reduction to the Newtonian mechanics, the gravity appears to exist apparently as an attractive force and if GR was free of such force, then this would seem to be a contradiction. We better say gravity does no longer have its classical meaning and rather it is now cast into a new form as the one we see in GR and is of course able to reveal itself as a force in some limited cases.

http://www.uoregon.edu/~struct/courseware/461/461_lectures/461_lecture4/461_lecture4.html says:
"A "force" is an action that changes, or tends to change, the state of motion of the body upon which it acts." In other words, a force does work, right? So, I'm wondering if the relevant question might be, "what has energy?"

Why are planets in orbit? Isn't it because of opposing forces? Isn't it similar to tying a ball at the end of a string and spinning it in a circle? Energy, thus force, is required to resist something, isn't it?

Why don't planets take a geodesic path right into the sun? Isn't it because they have their own agenda - their own energy - that is trying to go somewhere else but the gravity prevents them from leaving?

All these questions lie in the fact that you're a bit of a stranger to fresh arguments of GR! Gravity though has a force-like nature, is more efficiently replaced systematically by the curvature of spacetime and as soon as you find this as a really useful touchstone to measure the effects of gravitational fields on the fabric of spacetime, it turns out to be like an easy essay which is going to sit right with you line-by-line. But remember that sometimes we don't ask why; we just simply ask ourselves how are planets orbiting? This is because no one has any information as to what happened millions of years ago at the advent of planets and stars! This is what we see and GR Physics tries to find out what is behind all these motions!

AB

 

[Dale]

DaleSpam, you're suggesting that gravity can still be defined as a force

No, I am suggesting that it does not matter at this point and that you should be focusing on trying to understand the idea of worldlines and curved geometry.

Why don't planets take a geodesic path right into the sun? Isn't it because they have their own agenda - their own energy - that is trying to go somewhere else but the gravity prevents them from leaving?

This is a much more important question, and you have the tools to answer it. Remember the "local" definition of a geodesic? How can you apply that definition here? Keep in mind that we are talking about the geometry of spacetime and not just space.

 

[dx]

dx, you and Nabeshin seem to be taking similar, diplomatic approaches to the problem by trying to overlook semantics. I think this is a noble approach, but it also seems like a cop-out.

I don't see how its a cop-out. There are two ways that your question can be interpreted. One has to do with the difference between gravity and the other interactions, and this definitely is a meaningful question, since our description of gravity is distinctly different from our descriptions of the other three interactions. What you seem to worried about is the more naive question of "is gravity curvature of spacetime or is gravity a force?". This second question definitely does not make sense, and you don't even have to go to general relativity to see why; it can be illustrated in classical mechanics itself. Newton's description of motion involves the notion of 'force', and the phenomenon of gravity is represented in this description by the force F = GmM/r². But the actual content of Newton's theory is not the assertion "F = ma", with force, mass and acceleration defined seperately, but the assertion "the mathematical structure of second order differential equatoins applies to the phenomenon of motion". This mathematical structure can be 'viewed' in different ways, and some of these ways involve the idea of 'force' and some of them don't. For example, the lagrangian formulation of mechanics, which is mathematically isomorphic to Newton's viewpoint does not involve the idea of force. The dynamics in this viewpoint is represented by a function called the lagrangian, and gravity enteres in the form L =mv₁²/2 + Mv₂²/2 + GmM/|r₁ - r₂|. So it doesn't make sense to ask "is gravity a largangian or a force?", since they mean exactly the same thing. They are two ways of looking at the same mathematical structure, i.e. the structure of second order differential equations.

 

[Altabeh]

... which is mathematically isomorphic to Newton's viewpoint does not involve the idea of force. The dynamics in this viewpoint is represented by a function called the lagrangian, and gravity enteres in the form L =mv₁²/2 + Mv₂²/2 + GmM/|r₁ - r₂|. So it doesn't make sense to ask "is gravity a largangian or a force?", since they mean exactly the same thing. They are two ways of looking at the same mathematical structure, i.e. the structure of second order differential equations.

The Lagrangian approach leads to the idea of force for representing gravity via the fundamental formula

-\nabla L = \textbf{F}.

All is known to us from the Newtonian mechanics discussed in either way is that it has "gravity" defined as a force!

AB

 

[dx]

The Lagrangian approach leads to the idea of force for representing gravity via the fundamental formula

-\nabla L = \textbf{F}.

All is known to us from the Newtonian mechanics discussed in either way is that it has "gravity" defined as a force!

AB

That is simply a connection between the concepts of two equivalent descriptions of an underlying structure, which of course must exist (just like there is always a connection between the elements of two basis sets for the representaiton of a vector space). The point is that the notion of 'force' does not even have to be introduced. If mechanics were discovered historically in the lagrangian form, physics could have gone on just as well without ever using the word 'force' and simply talking about lagrangians.

 

[Altabeh]

That is simply a connection between the concepts of two equivalent descriptions of an underlying structure, which of course must exist (just like there is always a connection between the elements of two basis sets for the representaiton of a vector space). The point is that the notion of 'force' does not even have to be introduced. If mechanics were discovered historically in the lagrangian form, physics could have gone on just as well without ever using the word 'force' and simply talking about lagrangians.

Nevertheless there is such a word "force" and people have been keeping to fall into the habit of saying that since physics was born. The Lagrangian by itself can't be so much useful and that is the formulae like the one I gave, or the action formula, that let the theory blossom the applicative power of Lagrangians in physics!

AB

 

[dx]

The Lagrangian by itself can't be so much useful and that is the formulae like the one I gave, or the action formula, that let the theory blossom the applicative power of Lagrangians in physics!

AB

No, the formula that you gave really says nothing and is not needed in the application of lagrangians, let alone "let the theory blossom their applicative power". Also, the formula that you gave is actually wrong.

The fundamental equation of motion in the lagrangian formulation is (d/dt)(∂L/∂q') = ∂L/∂q. Only when we use the cartesian coordinates x, y, z can we think of ∂L/∂q as the vector quantity (F_x, F_y, F_z). For other generalized coordinates, ∂L/∂q does not even transform as a vector; it is a new type of object called a 1-form. So in fact the idea of force in the Newtonian sense has a limited domian of usefulness.

 

[Altabeh]

No, the formula that you gave really says nothing and is not needed in the application of lagrangians, let alone "let the theory blossom their applicative power". Also, the formula that you gave is actually wrong.

Please be careful when denying something that is already known to be true: I only made a typo and used a minus sign while this would have been a plus sign. This formula is not wrong at all and of course it is given in the Cartesian coordinates to lead to Newton's second law!

The fundamental equation of motion in the lagrangian formulation is (d/dt)(∂L/∂q') = ∂L/∂q. Only when we use the cartesian coordinates x, y, z can we think of ∂L/∂q as the vector quantity (F_x, F_y, F_z). For other generalized coordinates, ∂L/∂q does not even transform as a vector. It is a new type of object called a 1-form.

Who talked about "generalized coordinate system"? You and I, as I remember correctly, were talking about the Lagrangian approach to Newtonian mechanics and you gave in an early post the gravitational Lagrangian and I just said that you're wrong if you claim there is no such thing as "force" in the new approach! Again Lagrangian is just a key to many doors and until there we don't see a door, this key cannot come in handy! Period!

AB

 

[dx]

Please be careful when denying something that is already known to be true: I only made a typo and used a minus sign while this would have been a plus sign. This formula is not wrong at all and of course it is given in the Cartesian coordinates to lead to Newton's second law!

When I said it was wrong, what I meant was that it has a limited applicability, as I explained right after that statement.

You and I, as I remember correctly, were talking about the Lagrangian approach to Newtonian mechanics and you gave in an early post the gravitational Lagrangian and I just said that you're wrong if you claim there is no such thing as "force" in the new approach!

I don't know what you were talking about, but I was talking about the Lagrangian and the Newtonian approaches as two equivalent approaches to classical mechanics. One involves the concept of 'force', the other does not, and need not. The fact that 'force' and 'lagrangian' can be related, which is obvious, does not imply that the lagrangian formulation needs the idea of force.

 

[Altabeh]

When I said it was wrong, what I meant was that it has a limited applicability, as I explained right after that statement.

I really don't know what logic is behind the statement 'if a formula has "limited applicability" so it is worng'! Maybe the other users know!

I don't know what you were talking about, but I was talking about the Lagrangian and the Newtonian approaches as two equivalent approaches to classical mechanics. One involves the concept of 'force', the other does not, and need not. The fact that 'force' and 'lagrangian' can be related, which is obvious, does not imply that the lagrangian formulation needs the idea of force.

If you really don't, I don't see a reason to keep this going!

AB