Is it consistent to define the position of a particle in some frame as a vector or is just an informal representation? Velocity and acceleration can be added up and multiplied by real numbers and still have physical meaning (they live in tangent spaces). But what is the physical meaning of adding a position to another or, in relativity's domain, does adding two events have any sense? If this has no sense, then defining the position of a particle as a "vector living in a vector space" is surely wrong?
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$\begingroup$ Closely related/possible duplicate: physics.stackexchange.com/a/192887/226902 "Does spacetime position not form a four-vector?", see also: physics.stackexchange.com/q/156395/226902. Regarding affine Minkowski space: physics.stackexchange.com/a/279215/226902 "Four-vector addition in Minkowski space". $\endgroup$– QuilloCommented Jan 30, 2023 at 11:20
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3 Answers
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It depends on context:
In the context of affine spaces (such as, Newtonian mechanics or SR), positions are strictly speaking not vectors, but position differences/displacements are vectors, i.e. positions measured relative to some chosen origin/fiducial point are vectors.
In the context of manifolds (such as, GR or Newton-Cartan theory), positions are generally not vectors.
answered Jan 18, 2020 at 18:52
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$\begingroup$ Really cool answer. Equivalently, we may say that in SR the displacement is a vector because it belongs to the local tangent space to the "origin/fiducial point", which can be easily seen as a vector space (while the origin/fiducial point is a point of the manifold in GR). $\endgroup$– QuilloCommented Jan 30, 2023 at 11:17
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No, positions are not really vectors in a vector space. They are points in an affine space https://en.m.wikipedia.org/wiki/Affine_space . An affine space comes with a vector space though and you can use the vector space to parametrize the affine space nicely. This, is of course, the classical (and in fact, special relativistic I.e. no general relativity) description.
answered Jan 18, 2020 at 18:52
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Historically, vectors were introduced in geometry and physics (typically in mechanics)
Geometry means three dimensional space
A Euclidean vector, is thus an entity endowed with a magnitude (the length of the line segment (A, B)) and a direction (the direction from A to B). In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to scalars, which have no direction. For example, velocity, forces and acceleration are represented by vectors.
This includes defining the location of a particle in three dimensional space.
Actually the question should be :
In a general physical sense, can the th position of a particle really a vector?
as assigning numbers to particles is the process of modeling with algebra and mathematics.
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$\begingroup$ So define position as Vector makes sense only in a euclidian space, but not in more general spaces, like manifolds? $\endgroup$ Commented Jan 18, 2020 at 18:38
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$\begingroup$ Of course, as long as the "vectors" obey vector algebra, and form vector spaces. The question discusses position, which is either 3 vectors, or four vectors for relativity and general relativity $\endgroup$– anna vCommented Jan 18, 2020 at 18:45
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$\begingroup$ Position is not a vector in GR though. Strictly speaking it's not a vector in SR either. Rather, you specify a displacement vector from an arbitrary point you call "the origin". $\endgroup$ Commented Jan 21, 2020 at 23:50
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$\begingroup$ @user76284 three vectors are part of four vectors , so a four vector defines also the position in a given inertial frame, No? $\endgroup$– anna vCommented Jan 22, 2020 at 5:16