Action & Reaction: Equal Opposites

[WannabeNewton]

That's because you're writing the ordinary derivative d/dτ which is coordinate dependent, instead of the absolute derivative D/Dτ which is not. See the definition on my blog.

Yes but the absolute derivative just tells us that the proper acceleration vanishes identically; my point is that whether or not the coordinate acceleration vanishes depends on the coordinates chosen. The statement "the velocity of the object doesn't change" is not absolute if one is to interpret the statement as referring to the coordinate acceleration $\frac{du^{\mu}}{d\tau}$ as opposed to the absolute acceleration. A person standing on the ground of the Earth has all the right to say that the ball is accelerating downwards towards him if acceleration refers to coordinate acceleration; only in the coordinates obtained from a frame freely falling with the ball will the vanishing absolute acceleration agree with the vanishing coordinate acceleration. I have read your blog before by the way, and it is a brilliant blog (I also liked your blog on Fermi-Walker transport). Cheers!

EDIT: Just to clarify, I agree completely with Nugatory that the dust grains stuck on the ground and people standing on the ground have a non-vanishing absolute acceleration whereas the ball has a vanishing absolute acceleration but in what absolute sense are the dust grains stuck on the ground and people standing on the ground accelerating "upwards towards" the ball?

 

[A.T.]

To my small mind/brain, it is still not at all clear as to why the speed of the ball should increase on falling from the roof when there is nothing called as 'gravitational force?

Did you watch the video in post #20? It explains exactly that, without math.

 

[pervect]

General relativity is not based on Newton's laws (such as the law of action and reaction you cite numerous times) Insisting that it is or should be just isn't going to get you very far.

Space-time curvature is a more general concept than a "force", thus while forces can be explained in terms of particular sorts of space-time curvature (i.e. via Newton Cartan theory), space-time curvature cannot, in its full generality, be fully described only by forces. Basically, space itself is "curved" according to GR, so for instance the sum of the angles of a triangle will in general be different from 180 degrees. "Forces" cannot change the sum of the angles of a triangle - space time geometry can.

The first step to actually learning GR would probably be to learn how to do Newtonian physics without forces, using Lagrangian methods - via the principle of least action.

 

[BruceW]

yeah, and using non-cartesian reference frames, with a metric that is not just the identity (for example spherical polar coordinate system). The idea of fictitious forces is (in my opinion) probably the closest concept to general relativity, without actually being general relativity.

 

[DrGreg]

Einstein says the ground is accelerated upwards towards the ball! So there is no force acting on the ball, causing it to accelerate down. There is a force acting on the ground causing it to accelerate up.

Can you take pains to explain these lines, without any mathematics, of course?

Maybe the following diagram will help, along with the explanations that others have already given.

A spacetime diagram is just a fancy name for a distance-versus-time graph. When there's no gravity, freely-moving objects are represented by straight lines drawn on a flat sheet of paper (diagram A).

When there's gravity, freely-moving objects are represented by lines drawn as straight as possible on a curved surface (diagram C). In this diagram, the surface of the Earth is represented by the thicker blue line that has an arrow labelled "t" pointing along it. This line is a curved line. The two red lines represent two balls dropped one after the other from a roof. These lines are as straight as possible in the curved surface.

For more details see the post where this diagram came from.

 

[Naty1]

Nuveen:

Partly because I've been reading a lot about cosmological models, where time is effectively treated in the Newtonian way.

No, in the generally accepted FLRW [or FRW] cosmological model, space and time are NOT Newtonian. Their behavior is described via relativity.

 

[BruceW]

eh? It was me who said that, not Nuveen. And time is treated differently to space in the FLRW metric, in a very Newtonian way. Space is still curved. But time is like a separate parameter, for example in spherical universe, the size increases as some function of time. (reminds me lot of the Newtonian way of thinking). Of course, spacetime is still spacetime, but because of the assumptions on what makes up the universe, it seems like a 'Newtonian model' in a certain sense.

edit: to be clear, I am not literally saying that the FLRW metric uses Newtonian physics. I just mean that in several ways, it is similar to a Newtonian model.

 

[soothsayer]

To my small mind/brain, it is still not at all clear as to why the speed of the ball should increase on falling from the roof when there is nothing called as 'gravitational force?

There is a difference, in relativity, between coordinate acceleration, and proper acceleration.

Coordinate acceleration is the Newtonian concept of acceleration. It is the acceleration of an object as viewed from an observer, so, the falling of the ball from the roof towards the surface of the Earth, you see that as an acceleration, classically attributed to gravity.

Proper acceleration is the physical acceleration measured by an object. Imagine sitting in a car an accelerating, you feel a force pushing you against your seat--your inertia keeping you in place. It is an acceleration that you can measure within your reference frame. Note that while accelerating to the Earth, the ball has NO proper acceleration: it feels no force, it is simply in free fall, so while it appears to be going through coordinate acceleration, according to you who are standing on the roof, it is actually YOU who are experiencing proper acceleration. How? Think about your weight. You feel weighted down because the roof you are standing on is opposing your free fall through spacetime. In fact, you are not weighted because of gravity, you are weighted because you are opposing gravity.

So, in this example, the ball is undergoing coordinate acceleration relative to you. This is not caused by a force, but rather, by the ball following the geodesics of spacetime. Because it is following those geodesics, it has no proper acceleration. You are actually the one that is accelerating away from the ball, in a proper sense, due to force from the roof allowing you to oppose the geodesics of spacetime.

 

[BruceW]

I like that explanation. copy, and paste into blog. hehe, nah I wouldn't. It is a nice way to explain it though.

 

[Naty1]

And time is treated differently to space in the FLRW metric, in a very Newtonian way. Space is still curved. But time is like a separate parameter, for example in spherical universe, the size increases as some function of time. (reminds me lot of the Newtonian way of thinking). Of course, spacetime is still spacetime, but because of the assumptions on what makes up the universe, it seems like a 'Newtonian model' in a certain sense.

I have no idea what that means... but better a subject for a separate discussion if you see fit. Newtonian time is a parameter for sure, relativity treats time with sign opposite to that of space... Time is relative in relativity, absolute in Newtonian. I can agree it's like Newtonian locally where your wristwatch ticks with unchanging precision.

 

[soothsayer]

I have no idea what that means... but better a subject for a separate discussion if you see fit. Newtonian time is a parameter for sure, relativity treats time with sign opposite to that of space... Time is relative in relativity, absolute in Newtonian. I can agree it's like Newtonian locally where your wristwatch ticks with unchanging precision.

I think BruceW is basically right. The FLRW metric basically just looks at the spatial component, which contains a scale factor dependent on a time parameter. It is otherwise independent of time, because it doesn't really care about the relativity of time between different observers, does it? But yeah, better discussion for another place, especially seeing as the OP is still confused.

 

[Naty1]

I like that explanation. copy, and paste into blog. hehe, nah I wouldn't.

Don't be proud...when you a find a good explanation, use it and attribute it to the source.

 

[Naty1]

It is otherwise independent of time, because it doesn't really care about the relativity of time between different observers, does it?

Of course it does care ..it is CRITICAL..."At rest wrsp to the CMBR" is what sets those observers time equal...Otherwise, distant observers, at different relative velocities, all have different elapsed times.

 

[Naveen3456]

There is a difference, in relativity, between coordinate acceleration, and proper acceleration.

Coordinate acceleration is the Newtonian concept of acceleration. It is the acceleration of an object as viewed from an observer, so, the falling of the ball from the roof towards the surface of the Earth, you see that as an acceleration, classically attributed to gravity.

Proper acceleration is the physical acceleration measured by an object. Imagine sitting in a car an accelerating, you feel a force pushing you against your seat--your inertia keeping you in place. It is an acceleration that you can measure within your reference frame. Note that while accelerating to the Earth, the ball has NO proper acceleration: it feels no force, it is simply in free fall, so while it appears to be going through coordinate acceleration, according to you who are standing on the roof, it is actually YOU who are experiencing proper acceleration. How? Think about your weight. You feel weighted down because the roof you are standing on is opposing your free fall through spacetime. In fact, you are not weighted because of gravity, you are weighted because you are opposing gravity.

So, in this example, the ball is undergoing coordinate acceleration relative to you. This is not caused by a force, but rather, by the ball following the geodesics of spacetime. Because it is following those geodesics, it has no proper acceleration. You are actually the one that is accelerating away from the ball, in a proper sense, due to force from the roof allowing you to oppose the geodesics of spacetime.

Since you people are the 'physics persons' here, proof of burden lies on you. So, I will be persistent in my efforts at understanding this concept. if not, I will just submit to what you say.

My next question is that it is okay that I have been stopped by the roof, so cannot follow the geodesic. The ball is free to do so, so I see it moving with respect to me. but why does it accelerate? why doesn't it just continue in uniform motion?

Suppose in empty space I and a ball are both moving side by side in uniform motion (i.e. ball is at rest with respect to me). I suddenly stop, will I see the ball accelerating from me or just moving away from me at the same speed(uniform motion).

Now you can say you are talking about linear motion and the space is not curved here. I say what's the proof that the portion of space in which I and the ball are moving in a uniform motion is not a small part of a vast curve that is unnoticeable/unfeelable to both me and the ball. Just like we consider the surface of the Earth to be flat when moving in uniform motion over it.

 

[A.T.]

The ball is free to do so, so I see it moving with respect to me. but why does it accelerate? why doesn't it just continue in uniform motion?

The ball starts moving in space, because its geodesic world-line deviates from the initially purely temporal path in distorted spacetime. See video at 0:48

https://www.youtube.com/watch?v=DdC0QN6f3G4

Suppose in empty space I and a ball are both moving side by side in uniform motion (i.e. ball is at rest with respect to me). I suddenly stop, will I see the ball accelerating from me or just moving away from me at the same speed(uniform motion).

During the process of stopping you will see the ball accelerate. Just as you see it accelerate when standing on the roof. But in both cases it is you who undergoes proper acceleration. The ball just undergoes coordinate acceleration in your frame of reference.

In the video above the green apple represents you standing on the roof, or changing your velocity in space. The red apple represents a free falling ball.

 

[BruceW]

I think A.T. answered pretty well. But also, about this bit:

Now you can say you are talking about linear motion and the space is not curved here. I say what's the proof that the portion of space in which I and the ball are moving in a uniform motion is not a small part of a vast curve that is unnoticeable/unfeelable to both me and the ball. Just like we consider the surface of the Earth to be flat when moving in uniform motion over it.

Yes, that is possible. In this case, the curve is only very slight, so there will only be a very slight effect which is due to the curve. Also, it is good to ask questions. Maybe it would be more time-efficient to learn some general relativity first though. (not trying to be condescending, I still am learning general relativity, and I've only learned some of the basics).

 

[soothsayer]

My next question is that it is okay that I have been stopped by the roof, so cannot follow the geodesic. The ball is free to do so, so I see it moving with respect to me. but why does it accelerate? why doesn't it just continue in uniform motion?

You need to be able to look at the mathematics of general relativity in order to understand why spacetime curvature causes acceleration, but I'll try to explain it qualitatively.

Imagine sitting on a curved, 3D surface, the top of a hill perhaps. You know that the motion of a ball across that 3D surface is influenced by the curvature, but something must actually give that ball a push to get it to accelerate, and move across the surface following geodesics. You're wondering where that push comes from.

In a spacetime metric, we have three spatial components: x, y, z, or r, θ, ∅, etc. depending on how you wish to express your coordinat system. We also have a TIME component. Now, when we start taking derivatives of this 3+1 dimensional metric, as we must in order to solve the metric's geodesics equations, we will be taking second derivatives of these spatial variables with respect to our time variable. Now, hopefully you can tell me what it is called when you take a second time derivative of a position? That is EXACTLY what acceleration is.

You're right, if we simply curved 3D space, we wouldn't expect any acceleration along geodesics, but since we are curving 3 dimensions of space and one of time, we get time derivatives of spatial variables which give us velocity and acceleration. Not acceleration due to any force, but due to the fact that following a 4 dimensional geodesic demands it. This is in perfect keeping with the Newton's laws, because objects travel in straight lines, and a curved, accelerated path is what actually constitutes a straight line in curved 3+1 space.

Suppose in empty space I and a ball are both moving side by side in uniform motion (i.e. ball is at rest with respect to me). I suddenly stop, will I see the ball accelerating from me or just moving away from me at the same speed(uniform motion).

If by empty space you mean there is no gravity acting on you and the ball, then the ball would move away from you at a constant velocity, except for the interval where you are decelerating to a stop: during that time the ball will appear to be accelerating relative to you (again: not proper acceleration, coordinate acceleration. You are the one who will experience the proper acceleration, because there must be a force causing you to stop.)

Now you can say you are talking about linear motion and the space is not curved here. I say what's the proof that the portion of space in which I and the ball are moving in a uniform motion is not a small part of a vast curve that is unnoticeable/unfeelable to both me and the ball. Just like we consider the surface of the Earth to be flat when moving in uniform motion over it.

You're asking how we can really tell whether a region of spacetime is curved or not? In free fall, you can't determine whether you are accelerating, or just moving at constant velocity, since there are no inertial forces on you. There are, however, many tests you could do to determine if you are in a gravitational field: consider that the gravitational field is not constant: it has a gradient and changes at every point. If you had some test particles, you could identify the gravitational field lines. If you had sensitive enough equipment, you could determine the difference in gravitational force acting on your head and your feet (also known as tidal forces--these become very strong around massive bodies. This force gradient is what heats the interior of Jupiter's moons.)

Remember that Einstein's equivalence principle only applies locally in a gravitational field, not globally.

Hopefully this will put your doubts to rest, but if you're still skeptical, don't be afraid to ask away. You'll never learn anything in Physics by simply taking what you're taught at face value; I learned that too late in my college career. I would like to echo Bruce in suggesting that you make sure you are equipped with the physics/math knowledge to understand the answer to your question first. A lot of times in advanced physics, it is hard to water down an explanation without diluting the truth.

 

[soothsayer]

Don't be proud...when you a find a good explanation, use it and attribute it to the source.

Feel free to do so, Bruce

 

[Naty1]

but why does it accelerate? why doesn't it just continue in uniform motion?

There are few answers to 'why' in physics...In this case it IS uniform motion from a GR perspective. YOU have to think new basics in GR, just like you do in quantum mechanics. Stuff is not so simple and straightforward as is commonly perceived.

In this case, you have to think differently than the Newtonian perspective you express. That's just one view. In GR inertial motion, that is uniform motion, follows geodesics, that is, paths in which the free falling mass feels no forces in the frame of reference of the mass particle. "feels no forces" is like Newtonian physics, where inertial motion also is one without applied external forces.

YOU would be accelerating right now in the frame of Earth's surface except the chair you sit in is pushing up against you. Remove the chair, you feel no force as you fall...just like a cannon ball after it is fired...THAT moves along a curve, accelerates towards earth, while maintaining a steady velocity horizontally. Gravity is an pseudo force analogous to centripetal force; What you feel is the restraint against the pseudo force.

It was Einstein's ability to take on new perspectives that enabled him to develop SR and GR. So give it a try!

 

[WannabeNewton]

Naveen, in GR the object isn't accelerating if it is in free fall. The key point is that a freely falling particle is locally inertial; that is, locally it travels on what we normally think of as straight lines. Globally, the trajectory will deviate from straight line motion due to curvature. There is no physics here: in curved geometries the notion of not accelerating is traveling on curves which are as straight as possible i.e. curves which locally can be made to look like the usual straight lines; these curves are called geodesics. The physics comes in when we state that freely falling particles travel on these geodesics.

 

[BruceW]

Even if there is zero curvature, (i.e. no massive objects warping spacetime), then we can still have acceleration along geodesics. It just depends on what coordinate system you choose. For example, if you use yourself as the spatial origin of a coordinate system, drop a ball in free space, then get in a rocket and zoom off, then according to your coordinate system, the ball accelerates away from you. So the ball accelerates even though no forces are acting on it. This is another example of why it is not surprising that as soon as we allow non-inertial reference frames, it is kind of obvious that Newton's laws (as we once knew them) no longer apply.

edit: this post isn't really a reply to any other post, just I hope this might be useful to Naveen, to see the kind of crazy stuff that general relativity allows us to do.

 

[Naveen3456]

curves which locally can be made to look like the usual straight lines; these curves are called geodesics. The physics comes in when we state that freely falling particles travel on these geodesics.

I am getting something of this. it has been 2 months since I have been 'studying' relativity primarily from the Internet ( I was having holidays) and some 'awakening' has already dawned upon me. But clouds of doubt/curiosity keep on shadowing this awakening.

My next question. Can all curves be made to look like straight lines and then physics applied to them?

Consider a solid 'curved' particle of the size of an electron. can all this be done to this particle also. Why not?

 

[Naveen3456]

You're right, if we simply curved 3D space, we wouldn't expect any acceleration along geodesics, but since we are curving 3 dimensions of space and one of time, we get time derivatives of spatial variables which give us velocity and acceleration. Not acceleration due to any force, but due to the fact that following a 4 dimensional geodesic demands it. This is in perfect keeping with the Newton's laws, because objects travel in straight lines, and a curved, accelerated path is what actually constitutes a straight line in curved 3+1 space.
.

Does it mean there can be no motion without time?

Does it mean time has 'independent' existence and is just not the rate of 'change' of things..

I thought all the motion resulting from the big bang created our universe. It's just motion from moving sub-atomic particles to moving galaxies. 'Time' appears to be rate of change of all this.

I think now I have to spend many days studying about 'time' even.

By the way, who was he who said that ignorance is bliss. JUST JOKING:shy:

 

[mfb]

Does it mean there can be no motion without time?

I don't see how "motion" could be defined without time.

Does it mean time has 'independent' existence and is just not the rate of 'change' of things..

How do you mean that question?

I thought all the motion resulting from the big bang created our universe.

No.

'Time' appears to be rate of change of all this.

We define the length of our timescales (seconds, for examples) based on rates of change of something. I don't think you can say "time is the rate of change" of anything.

 

[WannabeNewton]

My next question. Can all curves be made to look like straight lines and then physics applied to them?

No, this will only work for geodesics; in particular, with regards to massive particles, it will only work for freely falling particles (and keep in mind they can be made to look like straight lines only locally, that is in a sufficiently small region of space-time). I'm not sure what you mean by "and then physics applied to them" however; the framework of GR works for any and all worldlines, not just geodesics (so for example the dust grains sitting on the surface of the Earth).

 

[soothsayer]

Does it mean there can be no motion without time?

If time doesn't pass, how could things change position? Even in Newtonian physics, acceleration is the second derivative of a spatial coordinate with respect to time...the difference only being that in Newton's world, the passing of time is absolute.

Does it mean time has 'independent' existence and is just not the rate of 'change' of things..

In the framework of general relativity, yes, that is essentially the case. Time is treated as a separate dimension from space, with some interesting mathematical treatment that differs from that of space because time is a negative term in our spacetime metrics.

I thought all the motion resulting from the big bang created our universe. It's just motion from moving sub-atomic particles to moving galaxies. 'Time' appears to be rate of change of all this.

In the Newtonian view of things, this is basically the case: time is just a parameter that we use to track the change of systems, there isn't NECESSARILY any physical realism added to time (though I think Newton believed that there should be). I'm not sure I understand what you mean about the big bang...

In relativity, we have time dilation, which demonstrates that time is not an absolute parameter, but rather a sort of dimension that an observer will travel through differently relative to other observers depending on the strength of the gravitational field they are in and the speed at which they are moving relative to that other observer.

This is about as abstract as I can go without saying things that are just meaningless. Questions like "what is time?" or "is time real?" are more or less outside of the scope of physics right now. Really, we just know how to treat time mathematically, and can compare our mathematical knowledge of it to other things we think we understand to try to draw some comparisons between time and other things that are a bit easier for us to grasp. Anything else is just semantics/speculation.

 

[soothsayer]

Regarding the Big Bang. No, it is not the source of motion in our universe, it is simply what expanded space at the beginning of the universe. When I push this pencil on my desk forward, it certainly does NOT move because of the big bang...

 

[Naty1]

Does it mean there can be no motion without time?

Time keeps everything from happening at once.

Space keeps everything from happening to me.

 

[Naveen3456]

Regarding the Big Bang. No, it is not the source of motion in our universe, it is simply what expanded space at the beginning of the universe. When I push this pencil on my desk forward, it certainly does NOT move because of the big bang...

In my opinion motion is related to big bang as follows:

1. Big bang happened, space expanded. This is motion.

2. First there was just energy. It was not static. This is motion

3. Next, sub atomic particles and light atoms formed. They were also moving (including electrons and other nuclear particles in a single atom). This is motion.

4. After that, particles started coming together and various structures formed. This is motion.

5. Then atoms on a particular planet called Earth, started coming together and formed unicellular organisms. This is motion.

6. Then complex beings like human beings formed when more cells gradually developed bigger structures. This is motion.

7. These cells in my brain indulge in synaptic firing and other 'processes' which lead to a decision being made by me. All these processes involve motion. This is motion.

So, IMHO, if you push a pencil on a desk, it seems to be due to big bang only.

 

[A.T.]

ISo, IMHO, if you push a pencil on a desk, it seems to be due to big bang only.

Everything is due to big bang.