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In the book General Physics by L.D.Landau, he mentioned the following in the first and second paragraph

The fundamental concept of mechanics is that of motion of a body with respect to other bodies. In the absence of such other bodies it is clearly impossible to speak of motion, which is always relative. Absolute motion of a body irrespective of other bodies has no meaning.

The relativity of motion arises from the relativity of the concept of space itself. We cannot speak of position in absolute space independently of bodies therein, but only of position relative to certain bodies.

I am not very sure what he means by "We cannot speak of position in absolute space independently of bodies therein, but only of position relative to certain bodies."

My current understanding is that, for an absolute space without respect to any body, we can take any value as the position of a point as we did not specify the origin. Only by referring to some bodies, we can then get a position for the point.

However, I am not very sure about my understanding and I do not know if it is correct. (I feel there is some logical gap there... Why can't we just take any random point in the absolute space as origin? Is it because there is no difference between any two points in the absolute space if we do not consider some objects?...)

These are my thoughts and I am a bit confused... So, how should we interpret this sentence? What is the meaning of relativity of space here?

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    $\begingroup$ If there were only 1 object in the Universe the only coordinate system possible would relate to it. Its position or motion would be meaningless. $\endgroup$ May 25 at 15:50
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    $\begingroup$ It's important to note that in our current theories, space is absolute. The motion of a single object is mathematically a well defined concept (with the usual relativity caveat); whether it makes sense physically is a different question, and not a settled one. $\endgroup$
    – Javier
    May 25 at 22:22
  • $\begingroup$ Draw a circle. Now describe where its center is without referring to the circle. $\endgroup$
    – Jens
    May 26 at 11:23
  • $\begingroup$ @Javier I disagree. General relativity is our best theory of the structure of space, and it is not at all clear in GR what should be meant by an object's being at rest in any absolute sense. Only in cases of very special symmetry (stationary spacetimes) does such a notion exist. Is this what you meant by "the usual relativity caveat"? If so, it seems that the relativity caveat is the entire claim. $\endgroup$
    – jawheele
    May 26 at 19:15
  • $\begingroup$ @jawheele I never said an object could be in absolute rest. I said that the motion of a single object is a well defined concept: a curve on a fixed background manifold. And there are some absolute things we can say, like whether this curve is a geodesic or not. $\endgroup$
    – Javier
    May 26 at 19:37
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Yes, you can take a random point in space as the origin of your coordinate system and a random orientation for an associated frame of reference, and then measure distances, velocities, etc. relative to that reference frame - but the point is that this is a random choice of reference frame. Whereas we expect physically meaningful quantities, such as the distance between two objects, or any laws that determine how those objects behave, to be independent of any random choices that we happen to make.

When Landau says we cannot "speak of" absolute position, absolute motion, etc., he means that we should not base physical laws on those attributes. Any candidate for a physical law that depends on our choice of coordinate system (or that depends on whether we measure distances in metres or feet, or what day of the week it is) cannot be correct.

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Yes, you can take a random point as your origin, but how do you know which random point you have picked? Imagine you are floating alone in empty space with nothing for a thousand km in any direction. If you pick a point somewhere around you at random to be your origin, how do you know where it is? The only way you can do it is to specify it in relation to something physical. You could place a pin at the spot you believe you have picked, but you would be faced with the challenge of placing it in such a way that you left it there motionless with respect to the point it was intended to mark. But how would you know you had? How would you know that the pin was not drifting away from your intended point? You couldn't know, of course, because a point in empty space is an utterly abstract notion. The only way in practice in which you can define an origin is by reference to some physical body.

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We can pick any point in space as an origin. However, physics doesn't care what choice we make (any choice of origin gives the same physics), since it turns out the long-range forces in physics (gravity, E&M) depend only on the relative distance between objects (in fact, since the forces depend on $r^{-2}$, not even the orientation matters).

Since any choice is valid, we remove the redundancy of description by removing the concept of "origin".

Mathematically, LL is describing an affine space, which are the collection of relative positions between objects. For example, the vector pointing from an object at $(0,0)$ to an object at $(1,1)$ would be the same vector pointing from an object at $(1,0)$ to $(2,1)$.

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Your understanding is basically correct. To answer your question: "Why can't we just take any random point in the absolute space as origin?", we really can. The more technical issue with that is that space itself is not an immutable set of dimensions. Space itself is not static. It is expanding, and general & special relativity introduce some other caveats that make this a hard task.

The concept that your textbook is introducing here is a more local understanding of space. If you've taken Linear Algebra or any Vector Calculus, you're probably familiar with the rigorous definition of vectors and linear transformations/coordinate transformations. Vectors can only be representative of a position when you have a defined coordinate system and an origin. What you're textbook is attempting to explain here is that there is no preferred frame of reference in our universe and that positions and velocities can only be measured with respect to other positions and velocities.

It is less of an esoteric statement than it might initially appear; it is simply a consequence of how we define vectors. If I didn't explain anything clearly, feel free to ask any follow-up questions.

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  • $\begingroup$ I was waiting for the first answer to include the term "relativity" as in general/special relativity, and here it is ;). I'd encourage you to make clear that Einstein's relativity has nothing to do with what OP asks - that just muddles the water, I think. $\endgroup$
    – AnoE
    May 26 at 8:58
  • $\begingroup$ Yeah, I suppose I did not make it extra clear that the caveats I mentioned are not particularly relevant to the question. I think the rest of the commenters have detailed what I said in less technical terms and without the use of vectors (as I'm not sure OP has a clear and somewhat rigorous understanding of them). Thanks for the input :) $\endgroup$
    – Azerack
    May 26 at 14:49
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I am not very sure what he means by "We cannot speak of position in absolute space independently of bodies therein, but only of position relative to certain bodies."

Let's say you are trying to describe your location right now to someone. How would you do that? It's impossible without describing your location relative to something else. We have addresses, but those are all well defined and relative to something. We have GPS coordinates which are relative to the Earth. They are latitude and longitude. Latitude is defined based on degrees from the equator which is based on the rotation of the Earth. Longitude is defined based on the prime meridian which is essentially arbitrary but defined by a point on the Earth.

The same holds true for any location in space. Just try to define where something is without referencing another object.

Why can't we just take any random point in the absolute space as origin?

How are you going to define that random point in 'the absolute space'? Say you want to describe the current position of Voyager 1. Choose an arbitrary point in space as the origin. How are you going to describe the location of that point? It's impossible without using other bodies as references because there is no absolute space. Let's try a few arbitrary points and describe the location of Voyager and it's movement:

  1. "5 meters from Voyager's antenna in the direction it is facing" - Voyager is not moving. It will always be in the same location relative to that point because the point is defined in relation to Voyager. If you say 'Wait a minute, Voyager moved, so the point is where it was before voyager moved!'. Well, Voyager moved relative to some things, but then you are really defining the point relative to those things. In essence you are defining the location of Voyager relative to those things with an offset.

  2. "The Sun" (or Earth) - this is the normal reference for objects near us because the gravity of the Sun dominates motion in it's vicinity. Voyager is about 19 billion km away and moving away at about 56,000 km/h.

  3. "Andromeda Galaxy" - Voyager is about 24,000,000,000,000,000,000 km away and moving towards Andromeda at about 400,000 km/h (along with the Sun, the Earth, and the rest of the Milky Way)

There is no absolute space so there is no way to define the location of your random point you want to use as the origin without describing it in relation to other bodies.

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