# Galilean relativity in terms of homogeneity

• I
• gionole
In summary: it would be hard to argue that without it, we would not be able to do the same things we do in inertial frames today.
gionole
TL;DR Summary
Landau's book
I have a question related to Landau's book. In that, he says:

If we were to choose an arbitrary frame of reference, space would be in- homogeneous and anisotropic. This means that, even if a body interacted with no other bodies, its various positions in space and its different orienta- tions would not be mechanically equivalent. The same would in general be true of time, which would likewise be inhomogeneous; that is, different in- stants would not be equivalent. Such properties of space and time would evidently complicate the description of mechanical phenomena. For example, a free body (i.e. one subject to no external action) could not remain at rest: if its velocity were zero at some instant, it would begin to move in some direc- tion at the next instant.

It is found, however, that a frame of reference can always be chosen in which space is homogeneous and isotropic and time is homogeneous. This is called an inertial frame. In particular, in such a frame a free body which is at rest at some instant remains always at rest.

As an example, I'd like to bring a car and the ball hung inside the car and we can look at it from 2 different frames of reference.

Frame of reference is me(I'm inside the car): If car moves with constant speed, nothing happens to the ball, but if car accelerates, ball starts to swing back or forward(depending on acceleration forward or backward). You could say that in acceleration mode, space inside the car is inhomogeneous, anistropic as ball behaves differently.

When we choose a frame or reference as the person outside the car, to him, wouldn't the ball inside car behave the same way as observed from my reference frame ? It's like things are kind of mixed up in my head and can't put them in order.

Would appreciate the complete, good explanation of Landau's thoughts(I think he is repeating Galillei's relativity, but still) in terms of my example.

gionole said:
Would appreciate the complete, good explanation of Landau's thoughts(I think he is repeating Galillei's relativity, but still) in terms of my example.
It seems a slightly complicated way of putting things. The idea is that the underlying space is the same everywhere. Space, of course, isn't a vacuum and is filled with galaxies, stars, planets and other smaller objects. So, it's more of a theoretical point that there is no inherent difference in the space itself.

Likewise with time, there is no inherent difference between time today and time in the past or future. Systems evolve (galaxies, stars, planets, living organisms), but time itself (in some fundamental way) does not change.

The irony is that with General Relativity and modern Cosmology, this is not quite true. The expanding universe means that spacetime itself does change. But, while you are studying Newtonian physics, space and time and homogeneous and isotropic.

In Newtonian physics, we have the concept of an inertial reference frame: where objects subject to no external force remain at rest or moving with constant velocity. These are "special" frames in which the laws of Newtonian mechanics apply directly: in particular, Newton's second law.

In any other frame (e.g. the frame of an accelerating car), you cannot simply apply Newton's second law. To do Newtonian mechanics, you must include so-called inertial or fictitious forces (to take account of the acceleration of the reference frame itself).

vanhees71
@PeroK

Newton's second law is F = ma.

These are "special" frames in which the laws of Newtonian mechanics apply directly: in particular, Newton's second law.
If we have inertial frame, then acceleration is 0, F = 0, so how does 2nd law apply ? It's pointless, no ?

In any other frame (e.g. the frame of an accelerating car), you cannot simply apply Newton's second law. To do Newtonian mechanics, you must include so-called inertial or fictitious forces (to take account of the acceleration of the reference frame itself).
If we imagine the ball hung in a car and write down F = ma (m is the mass of the ball). Are you saying that we can't apply F=ma because we don't know the force acting on it since it's fictitious ? If we look at it from a different frame(outside the car), how can we apply it then ?

gionole said:
@PeroK

Newton's second law is F = ma.
Yes.
gionole said:
If we have inertial frame, then acceleration is 0, F = 0, so how does 2nd law apply ? It's pointless, no ?
There's often a debate about whether Newton's first law is necessary. Isn't is just a special case of the second law?

Newton's first law can be interpreted, as I mentioned above, as implying the existence of inertial reference frames.

My opinion, from looking at Newton's original work, is that the first law was so radical that he needed to emphasise it. It went against previous physical theory, so he wanted to spell it out before stating his second law.
gionole said:
If we imagine the ball hung in a car and write down F = ma (m is the mass of the ball). Are you saying that we can't apply F=ma because we don't know the force acting on it since it's fictitious ? If we look at it from a different frame(outside the car), how can we apply it then ?
You're confusing spatial locations with frames. The car's frame extends across all of space (and time). In the car's frame a lampost accelerates, but there is no force on the lampost. In that frame, you need to add a fictitious force to explain the acceleration of objects not physically bound to the car. A ball in the car might be another example. You have to be clear about what "real" forces are acting on something - and whether the real forces explain the motion. If they don't, then you must be using a non-inertial frame.

Note that I used the word "using", not "in" a frame. Everything is in all frames. A reference frame is not a thing, but a system of coordinates that you use to make measurements (of time and position).

Dale
@PeroK
You're confusing spatial locations with frames. The car's frame extends across all of space (and time). In the car's frame a lampost accelerates, but there is no force on the lampost. In that frame, you need to add a fictitious force to explain the acceleration of objects not physically bound to the car. A ball in the car might be another example. You have to be clear about what "real" forces are acting on something - and whether the real forces explain the motion. If they don't, then you must be using a non-inertial frame.

Q1: I don't think I'm. I said that in a car's frame, lampost accelerates for sure, but in order to write F=ma for the lampost, we need to know the force which we don't know that easily since the force is fictitious. So here, we don't know F and we don't know a(acceleration) as well, right ? but wouldn't lampost accelerate with the same magnitude of the car's acceleration but in opposite direction ? What is the problem that we're trying to solve here and we can't from the car's frame ? I think this is bugging me. Would you care to explain it a little bit further what the problem is here ? Is the exercise to define the motion of lampost and we can't do it because we don't know F ?

Q2: What I meant by "outside the car's frame" is that how is it a better frame(than car's frame) to deal with the problem in my Q1 ?

gionole said:
If we have inertial frame, then acceleration is 0, F = 0, so how does 2nd law apply ? It's pointless, no ?
No. The inertial frame is a mathematical device. Objects can accelerate in that inertial frame.

vanhees71 and Dale
gionole said:
If we have inertial frame, then acceleration is 0, F = 0, so how does 2nd law apply ? It's pointless, no ?
An inertial frame does not mean that acceleration is 0. An inertial frame means that any acceleration of an object is the result of a real force. I.e. it means there are no fictitious forces.

gionole said:
When we choose a frame or reference as the person outside the car, to him, wouldn't the ball inside car behave the same way as observed from my reference frame ?
No. The biggest difference is that in the non-inertial frame the point where the ball is attached is stationary, while in the inertial frame that point is accelerating. In the non-inertial frame the presence of the fictitious force is required in order to explain the motion of the ball. In the inertial frame the real forces are sufficient.

vanhees71 and PeroK
gionole said:
If we have inertial frame, then acceleration is 0, F = 0
You are confusing the acceleration of a reference frame with the acceleration of a mass.

vanhees71 and Dale
gionole said:
@PeroKQ1: I don't think I'm. I said that in a car's frame, lampost accelerates for sure, but in order to write F=ma for the lampost, we need to know the force which we don't know that easily since the force is fictitious. So here, we don't know F and we don't know a(acceleration) as well, right ?
We never know force, mass or acceleration unless we measure them.
gionole said:
but wouldn't lampost accelerate with the same magnitude of the car's acceleration but in opposite direction ?
Yes.
gionole said:
What is the problem that we're trying to solve here and we can't from the car's frame ? I think this is bugging me. Would you care to explain it a little bit further what the problem is here ? Is the exercise to define the motion of lampost and we can't do it because we don't know F ?
The issue is that for the lampost the real force ##F = 0##, but the acceleration is non-zero. Newton's second law is not directly obeyed in a non-inertial reference frame.
gionole said:
Q2: What I meant by "outside the car's frame" is that how is it a better frame(than car's frame) to deal with the problem in my Q1 ?
You can solve the same problem using difference frames of reference. If you choose a non-inertial frame you have to include the relevant fictious force(s).

vanhees71
Dale said:
An inertial frame does not mean that acceleration is 0. An inertial frame means that any acceleration of an object is the result of a real force. I.e. it means there are no fictitious forces.

No. The biggest difference is that in the non-inertial frame the point where the ball is attached is stationary, while in the inertial frame that point is accelerating. In the non-inertial frame the presence of the fictitious force is required in order to explain the motion of the ball. In the inertial frame the real forces are sufficient.
@Dale

As far as I understood, car frame reference is the non-inertial frame because lampost is accelerating, but no real force is acting on it. For the outside person, reference frame is inertial as you say and how does the person explain the motion of the lampost ? Since it's inertial, he should explain it caused by real force. What's the "real force" then ?

gionole said:
@Dale

As far as I understood, car frame reference is the non-inertial frame because lampost is accelerating, but no real force is acting on it. For the outside person, reference frame is inertial as you say and how does the person explain the motion of the lampost ? Since it's inertial, he should explain it caused by real force. What's the "real force" then ?
In the ground frame of reference, the lampost is not accelerating.

PeroK said:
In the ground frame of reference, the lampost is not accelerating.
I mean how ? I stand on the ground, I look at the car moving, and car accelerated which caused the acceleration of the lampost as it moved backward. Can you explain why I'd not see the lampost accelerating ?

gionole said:
I mean how ? I stand on the ground, I look at the car moving, and car accelerated which caused the acceleration of the lampost as it moved backward. Can you explain why I'd not see the lampost accelerating ?
Because it's fixed to the ground. It can't even move in the ground frame!

Ah, by lampost, we mean different things. Let me re-repeat my question that I asked to Dale above. I meant ball hung in a car instead of lampost.

Dale said:
An inertial frame does not mean that acceleration is 0. An inertial frame means that any acceleration of an object is the result of a real force. I.e. it means there are no fictitious forces.

No. The biggest difference is that in the non-inertial frame the point where the ball is attached is stationary, while in the inertial frame that point is accelerating. In the non-inertial frame the presence of the fictitious force is required in order to explain the motion of the ball. In the inertial frame the real forces are sufficient.

As far as I understood, car frame reference is the non-inertial frame because ball is accelerating, but no real force is acting on it. For the outside person, reference frame is inertial as you say and how does the person explain the motion of the ball ? Since it's inertial, he should explain it caused by real force. What's the "real force" then ?

gionole said:
I stand on the ground, I look at the car moving, and car accelerated which caused the acceleration of the lampost as it moved backward.
The acceleration of the lampost is relative to the car.

gionole said:
As far as I understood, car frame reference is the non-inertial frame because ball is accelerating, but no real force is acting on it. For the outside person, reference frame is inertial as you say and how does the person explain the motion of the ball ? Since it's inertial, he should explain it caused by real force. What's the "real force" then ?

What is the motion of the ball a) in the car frame and b) in the ground frame?

PeroK said:

What is the motion of the ball a) in the car frame and b) in the ground frame?
Well, by reading all of your answers, I understood that the inertial frame is when acceleration of the object can only be explained by "real force".

In the car frame, ball that is hung, accelerates because the car accelerates, but this acceleration can not be explained by real force as nothing external acted on the ball - i.e we got fictitious force.

Now, if you agree with me till now, then what do we call "the outside frame of reference"(imagine observer looks at car movement from outside - I think we can call this ground frame) ? We call it inertial frame(I derived this from Dale's answer), but we know in inertial frame, acceleration must be explained by the real force, not fictitious. So from this inertial frame, with what "real force" do you explain the ball's movement inside the car ?

gionole said:
In the car frame, ball that is hung, accelerates because the car accelerates, but this acceleration can not be explained by real force as nothing external acted on the ball - i.e we got fictitious force.
The ball is accelerated by the string - that's a real force. If it wasn't attached to something inside the car, then it wouldn't move until something inside the car hit it.

PeroK said:
The ball is accelerated by the string - that's a real force. If it wasn't attached to something inside the car, then it wouldn't move until something inside the car hit it.
Correct, but the real question is: why didn't we say that from car frame reference that ball was accelerated by the string and there must have been fictitious force?

gionole said:
Correct, but the real question is: why didn't we say that from car frame reference that ball was accelerated by the string and there must have been fictitious force?
I think you need to draw a diagram of this scenario with the real forces marked.

If the ball is attached to the car, then in the car frame it is not accelerating. But, there is a measurable real force on it. And, again, Newton's second law is not obeyed: force but not acceleration. So, we add a fictitious force to the ball to cancel out the real force and we have restored Newton's second law: net force is zero and acceleration is zero.

Dale
Basically, from car frame, we say fictitious force happened, and acceleration of the ball was caused by that fictitious force(hence non-inertial frame), while from the road reference, acceleration of the ball was caused by real force(string) hence inertial frame.

Why do we say that from car frame reference ball was accelerated by fictitious force, but from road frame, by string force ? why couldn't we even have said ball was accelerated by string force even in the car frame ?

@PeroK

gionole said:
Basically, from car frame, we say fictitious force happened, and acceleration of the ball was caused by that fictitious force(hence non-inertial frame), while from the road reference, acceleration of the ball was caused by real force(string) hence inertial frame.

Why do we say that from car frame reference ball was accelerated by fictitious force, but from road frame, by string force ? why couldn't we even have said ball was accelerated by string force even in the car frame ?

@PeroK
You're replying to these posts so quickly that you are clearly taking only a few seconds to think. I suggest you get a cup of coffee and have a good think about everything in this scenario.

Dale
I mostly understand what you mean and I will definitely start thinking, but before I do, could you quickly mention one thing ? you said: "If the ball is attached to the car, then in the car frame it is not accelerating", why is that ? If I'm in a car and car accelerated, ball will definitely move to the opposite direction. Why is this not an acceleration of the ball to me(person sitting in a car) ? @PeroK

gionole said:
If we have inertial frame, then acceleration is 0, F = 0, so how does 2nd law apply ? It's pointless, no ?
Not at all. We can use an inertial frame to analyze the motion of objects, including objects that are subject to forces, and Newton's second law is essential for that. You may be conflating the notions of "inertial frame" and "inertial motion" - objects may be moving inertially or not no matter what frame we are using to describe that motion and no matter whether that frame is inertial or not.

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gionole said:
I stand on the ground, I look at the car moving, and car accelerated which caused the acceleration of the lampost as it moved backward.
Do you really believe that a lamp post moves to a new location every time a car passes by!? The lamp posts in my neighborhood have remained in the same place since they were constructed decades ago.

Dale and PeroK
gionole said:
I mostly understand what you mean and I will definitely start thinking, but before I do, could you quickly mention one thing ? you said: "If the ball is attached to the car, then in the car frame it is not accelerating", why is that ? If I'm in a car and car accelerated, ball will definitely move to the opposite direction. Why is this not an acceleration of the ball to me(person sitting in a car) ? @PeroK
The ball accelerates initially until the string gets a chance to pull it along with the car. That's an irrelevant complication. It just means the real force on the ball acts slightly after the real force on the car.

gionole said:
"If the ball is attached to the car, then in the car frame it is not accelerating", why is that ? If I'm in a car and car accelerated, ball will definitely move to the opposite direction.
If the ball is "attached" to the car we take that to mean the ball doesn't move relative to the car.

PeroK said:
The ball accelerates initially until the string gets a chance to pull it along with the car. That's an irrelevant complication. It just means the real force on the ball acts slightly after the real force on the car.
To add to this. If you want to set up a scenario where the ball is essentially a pendulum inside an accelerating car, then that's a little more complicated. The same principles apply. In an inertial frame, the motion of the ball is explained by the real forces from gravity and the string. In the frame accelerating with the car, you have to add the fictitious force to explain the ball's motion relative to the car.

Given your conceptual difficulties with things like lamposts bolted to the ground, I would leave the more complex scenarios until you have grasped the basics.

gionole said:
In the car frame, ball that is hung, accelerates because the car accelerates, but this acceleration can not be explained by real force as nothing external acted on the ball - i.e we got fictitious force.
In the car frame (assuming uniformly accelerating car) the ball does not accelerate. It hangs at rest at an angle. The tension should make it accelerate horizontally, but it does not. So the fictitious force must be pulling it backwards to prevent it from accelerating.

gionole said:
in inertial frame, acceleration must be explained by the real force, not fictitious. So from this inertial frame, with what "real force" do you explain the ball's movement inside the car ?
In the inertial frame the ball is accelerating horizontally. The real force is the tension acting on the ball.

gionole said:
Correct, but the real question is: why didn't we say that from car frame reference that ball was accelerated by the string and there must have been fictitious force?
In the car's frame the ball is not accelerating.

gionole said:
"If the ball is attached to the car, then in the car frame it is not accelerating", why is that ? If I'm in a car and car accelerated, ball will definitely move to the opposite direction.
One problem is that you are not clear about the setup you are proposing. I am assuming for simplicity that the car is accelerating uniformly. You have not stated clearly what acceleration profile you are considering.

In a uniformly accelerating car, in the car's frame, a ball hanging does not accelerate. It simply hangs there at an angle from the vertical. The ball is hanging from a pivot point, that pivot point is attached to the car so it does not move in the car's frame, so the distance to the pivot point does not change and in the case of uniform acceleration the angle does not change either. The effect of the fictitious force is to allow the ball to not accelerate while the tension is pulling it forward.

The rest frame of a non-uniformly accelerating car is MUCH more complicated to analyze. I think you should avoid it for now. I would not participate in that discussion as it is more effort than it is worth.

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It seems like something is confusing to me, but I don't know what. If you can bear with me, I will explain how I understand the whole situation.

Car is moving with constant velocity 20m/s and then accelerates in uniform mode.

From ground frame: everything in the car was moving with the same 20m/s velocity and when car accelerated, the following happened: ceiling accelerated instantly, which caused the forward force(the force in the same direction as the car) to act on the upper point on the string, but force transfer through the string is not instantenous and there's a delay because of which the following occurs: the car moves with updated velocity, but ball itself still moves with old velocity(force didn't arrive yet through the end of the string where the ball is attached), so we got a difference in velocities - car moves faster than the ball which ends up in swing mode of the ball backwards. In reality, it's not truly that ball moves backward, it's just our perception, but in reality, it's just car moves faster than ball and thats why swinging perception of the ball occurs. We say this obeys in newton's law, because ball actually continued motion with the old velocity(if no force acted on it, object continues moving in the same speed and it's true due to force delay).

From the car frame: I got a problem here. I sit in a car. I understand that to me, when car moves with constant speed at first, everything is stationary - i look at ball or anything for that matter, and nothing moves. Now acceleration happened, but to me, ceiling is still stationary - to me, car/ceiling still feel like stationary. Now, if to me, if i feel that everything is still stationary, why did the ball swing back ? I think this is the scenario we try to explain and can't explain with the same logic as we did in ground frame. Since ceiling to me didn't accelerate, it couldn't have caused force in the tension the same way as in ground frame, but ball still swang backwards, so if no force acted on it, why did it move backwards ? It seems like newton's law broke - if no force, it should have stayed at rest or moving with the same speed, but the ball accelerated. The way you explain the swing of ball backwards is(i.e it's acceleration - because to me, it's not stationary during its backwards movement), some fictitious force must have happened.

Where would you say I'm wrong ? Would appreciate to point out the exact things that I wrongly said.

Thanks already so much for bearing with me. Physics in terms of intuitivity is lot harder than I imagined.

gionole said:
From the car frame: I got a problem here. I sit in a car. I understand that to me, when car moves with constant speed at first, everything is stationary - i look at ball or anything for that matter, and nothing moves. Now acceleration happened, but to me, ceiling is still stationary - to me, car/ceiling still feel like stationary. Now, if to me, if i feel that everything is still stationary, why did the ball swing back ? I think this is the scenario we try to explain and can't explain with the same logic as we did in ground frame. Since ceiling to me didn't accelerate, it couldn't have caused force in the tension the same way as in ground frame,
The ceiling didn't accelerate in the car frame, but the ball did. I don't think it's a good idea to start talking about "cause and effect" in a non-inertial frame, because it creates confusion.
gionole said:
but ball still swang backwards, so if no force acted on it, why did it move backwards ?
It moved backwards because of the fictitious force on it. The same fictitious force that applied to everything else in the car and cancelled out the real "accelerating" force.
gionole said:
It seems like newton's law broke - if no force, it should have stayed at rest or moving with the same speed, but the ball accelerated. The way you explain the swing of ball backwards is(i.e it's acceleration - because to me, it's not stationary during its backwards movement), some fictitious force must have happened.
The fictitious force applies as soon as you use the accelerating reference frame.
gionole said:
Where would you say I'm wrong ? Would appreciate to point out the exact things that I wrongly said.

Thanks already so much for bearing with me. Physics in terms of intuitivity is lot harder than I imagined.
With non-inertial frames you need to rely more on solid technique and mathematics. Your inituition is letting you down and you are drifting from one misconception to another. I suggest you start using free-body diagrams. In an non-inertial frame you add the fictitious force to everything.

Note that, in fact, you can combine the fictitious force with the "real" gravity to get an overall virtual-gravity in the non-inertial frame. In the car frame this acts at an angle, down and backwards. The ball, therefore, exhibits the motion of a simple pendulum under this virtual gravitational force.

@PeroK

Q1. Was everything correct in ground frame analysis of mine ?

Q2:
The ceiling didn't accelerate in the car frame, but the ball did. I don't think it's a good idea to start talking about "cause and effect" in a non-inertial frame, because it creates confusion.
Isn't this what I said ? Could you re-read my "car frame reference" analysis ?

Q3:
It moved backwards because of the fictitious force on it. The same fictitious force that applied to everything else in the car and cancelled out the real "accelerating" force.
I asked the question: "why ball moved backwards" to convey the problem, but I answer it by myself in there. I think re-reading my analysis would be so appreciated.

gionole said:
Car is moving with constant velocity 20m/s and then accelerates in uniform mode.
This is precisely the non-uniform acceleration that I wished to avoid. Changing acceleration from 0 to a constant is non-uniform acceleration. I would do this as two separate scenarios

1) Car is moving with constant velocity
2) Car is accelerating uniformly

I am not getting involved in this one as it is, but good luck maybe others like @PeroK can help. There are too many complications for having a non-uniform acceleration that it is not worth it for me.

Dale said:
This is precisely the non-uniform acceleration that I wished to avoid. Changing acceleration from 0 to a constant is non-uniform acceleration. I would do this as two separate scenarios

1) Car is moving with constant velocity
2) Car is accelerating uniformly

I am not getting involved in this one as it is, but good luck maybe others like @PeroK can help. There are too many complications for having a non-uniform acceleration that it is not worth it for me.
hey Dale. You got it wrong. I didn't mean to bring non-uniform acceleration. The reason I brought 20m/s was I wanted to convey the idea that before accelerating, with constant speed of car, ball also was moving with 20m/s and when acceleration happened, my analysis of why ball didn't instantly receive the force due to force transfer delay caused the ball to continue moving with 20m/s and car with higher speed which caused the perception of the ball moving backwards. I don't want to complicate this even more, believe me. If 1) car is moving with constant velocity, I don't know what we need to discuss at all. I want to discuss uniform acceleration. I know you mean when car has a = 2 all the time which I agree, but as I. said I only brought 20m/s in the beginning to convey the speed difference in the ball/car due to delay. with uniform acceleration, my analysis should still be right as ball receives the updated speed at a delay due to force delay while car instantly increases speed by 2 hence swinging.

Having said that, would it be hard for you to let me know in your opinion what I've explained wrongly in my above analysis ? @Dale

gionole said:
I didn't mean to bring non-uniform acceleration. The reason I brought 20m/s was I wanted to convey the idea that before accelerating, with constant speed of car, ball also was moving with 20m/s and when acceleration happened,
When there is a "before accelerating" and a "when acceleration happened" then the acceleration is non-uniform. I don't want to deal with the transient mess of the swinging ball. Good luck.

gionole said:
would it be hard for you to let me know in your opinion what I've explained wrongly in my above analysis ?
Yes. That is why I don't want to do non-uniform acceleration. It is MUCH harder and provides no instructional benefit.

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