Reference_frames

Reference frames

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See also: Inertial frame

A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position and the orientation of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer. Or it may refer to both an observational reference frame and an attached coordinate system, as a unit.

1 Different aspects of "frame of reference"
2 Coordinate systems
3 Observational frames of reference
4 Particular frames of reference in common use
- 4.1 In fields other than Physics
5 See also
6 Footnotes

Different aspects of "frame of reference"

The need to distinguish between the various meanings of "frame of reference" has led to a variety of terms. For example, sometimes the type of coordinate system is attached as a modifier, as in Cartesian frame of reference. Sometimes the state of motion is emphasized, as in rotating frame of reference.

In this article the term observational frame of reference is used when emphasis is upon the state of motion rather than the coordinates. In this sense, an observational frame of reference allows study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. On the other hand, a coordinate system may be employed for many purposes where the state of motion is not the primary concern. For example, a coordinate system may be adopted to take advantage of the symmetry of a system. In a still broader perspective, of course, the formulation of many problems in physics employs generalized coordinates, normal modes or eigenvectors, which are only indirectly related to space and time. It seems useful to divorce the two aspects of a reference frame for the discussion below.

We therefore take the view that observational frames of reference and coordinate systems are independent concepts: namely, an observational frame (such as an inertial frame or non-inertial frame of reference) is a physical concept, while a coordinate system is a mathematical one, amounting to a choice of language used to describe observations.^[1] Consequently, an observer in an observational frame of reference can choose to employ any coordinate system (Cartesian, polar, curvilinear, generalized, …) to describe observations as seen from that frame of reference. A change in the choice of this coordinate system does not change an observer's state of motion, and so does not entail a change in the observer's observational frame of reference, and vice versa. This viewpoint can be found elsewhere as well.^[2]

Here is a quotation applicable to moving observational frames in a Euclidean three-space: ^[3]

We first introduce the notion of reference frame, itself related to the idea of observer: the reference frame is, in some sense, the "Euclidean space carried by the observer". Let us give a more mathematical definition:… the reference frame is... the set of all points in the Euclidean space with the rigid body motion of the observer. The frame, denoted $\mathfrak{R}$ , is said to move with the observer.… The spatial positions of particles are labelled relative to a frame $\mathfrak{R}$ by establishing a coordinate system R with origin O.… In a frame $\mathfrak{R}$ , coordinates are changed from R to R' by carrying out, at each instant of time, the same coordinate transformation on the components of intrinsic objects (vectors and tensors) introduced to represent physical quantities in this frame.

– Jean Salençon, Stephen Lyle Handbook of Continuum Mechanics: General Concepts, Thermoelasticity p. 9

and this:^[4]

As noted by Brillouin, a distinction between mathematical sets of coordinates and physical frames of reference must be made. The ignorance of such distinction is the source of much confusion… the dependent functions such as velocity for example, are measured with respect to a physical reference frame, but one is free to choose any mathematical coordinate system in which the equations are specified.

– P. Cornille in Essays on the Formal Aspects of Electromagnetic Theory p. 149 referring to L. Brillouin Relativity Reexamined

and this:^[5]

The idea of a reference frame is really quite different from that of a coordinate system. Frames differ just when they define different spaces (sets of rest points) or times (sets of simultaneous events). So the ideas of a space, a time, of rest and simultaneity, go inextricably together with that of frame. However, a mere shift of origin, or a purely spatial rotation of space coordinates results in a new coordinate system. So frames correspond at best to classes of coordinate systems.

– Graham Nerlich: What Spacetime Explains, p. 64

See also the discussion in Brading and Castellani.^[6]

Coordinate systems

Main article: Coordinate systems

See also: Generalized coordinates

Although the term "coordinate system" is often used (particularly by physicists) in a nontechnical sense, the term "coordinate system" does have a rather precise meaning in mathematics, and sometimes that is what the physicist means as well.

A coordinate system in mathematics is a facet of geometry or of algebra,^[7]^[8] in particular, a property of manifolds (for example, in physics, configuration spaces).^[9] The coordinates of a point r in an n-dimensional space are simply an ordered set of n numbers:^[10]^[11]

$\mathbf{r} =[x^1,\ x^2,\ \dots\ , x^n] \ .$

In a general Banach space, these numbers could be (for example) coefficients in a functional expansion like a Fourier series. In a physical problem, they could be normal mode amplitudes. In a robot design, they could be angles of relative rotations, linear displacements, or deformations of joints. Here we will suppose these coordinates can be related to a Cartesian coordinate system by a set of functions:

$x^j = x^j (x,\ y,\ z,\ \dots)\ ,$ $j = 1, \ \dots \ , \ n\$

where x, y, z, etc. are the n Cartesian coordinates of the point. Given these functions, coordinate surfaces are defined by the relations:

$x^j (x, y, z, \dots) = \mathrm{constant}\ ,$ $j = 1, \ \dots \ , \ n\ .$

The intersection of these surfaces define coordinate lines. At any selected point, tangents to the intersecting coordinate lines at that point define a set of basis vectors {e₁, e₂, …, e_n} at that point. That is:^[12]

$\mathbf{e}_i(\mathbf{r}) =\lim_{\epsilon \rightarrow 0} \frac{\mathbf{r}\left(x^1,\ \dots,\ x^i+\epsilon,\ \dots ,\ x^n \right) - \mathbf{r}\left(x^1,\ \dots,\ x^i,\ \dots ,\ x^n \right)}{\epsilon }\ ,$

which can be normalized to be of unit length. For more detail see curvilinear coordinates.

Coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system.^[13] If the basis vectors are orthogonal at every point, the coordinate system is an orthogonal coordinate system.

An important aspect of a coordinate system is its metric g_ik, which determines the arc length ds in the coordinate system in terms of its coordinates:^[14]

$(ds)^2 = g_{ik}\ dx^i\ dx^k \ ,$

where repeated indices are summed over.

As is apparent from these remarks, a coordinate system is a mathematical construct, part of an axiomatic system. There is no necessary connection between coordinate systems and physical motion. However, coordinate systems can include time as a coordinate, and can be used to describe motion. Thus, Lorentz transformations and Galilean transformations may be viewed as coordinate transformations.

General and specific topics of coordinate systems can be pursued following the See also links below.

Observational frames of reference

Main article: Inertial frame of reference

An observational frame of reference, often referred to as a physical frame of reference, a frame of reference, or simply a frame, is a physical concept related to an observer and the observer's state of motion. Here we adopt the view expressed by Kumar: an observational frame of reference is characterized only by its state of motion. ^[15] However, there is lack of unanimity on this point. In special relativity, the distinction is sometimes made between an observer and a frame. According to this view, a frame is an observer plus a coordinate lattice constructed to be an orthonormal right-handed set of spacelike vectors perpendicular to a timelike vector. See Doran.^[16] This restricted view is not used here, and is not universally adopted even in discussions of relativity.^[17]^[18] In general relativity the use of general coordinate systems is common (see, for example, the Schwarzchild solution for the gravitational field outside an isolated sphere^[19]).

There are two types of observational reference frame: inertial and non-inertial. An inertial frame of reference is defined as one in which all laws of physics take on their simplest form. In special relativity these frames are related by Lorentz transformations. In Newtonian mechanics, a more restricted definition requires only that Newton's first law holds true; that is, a Newtonian inertial frame is one in which a free particle travels in a straight line at constant speed, or is at rest. These frames are related by Galilean transformations.

In contrast, a non-inertial frame of reference is one in which fictitious forces must be invoked to explain observations. An example is an observational frame of reference centered at a point on the earth's surface. This frame of reference orbits around the center of the earth, which introduces a fictitious force known as the Coriolis force.

Simple example

Figure 1: Two cars moving at different but constant velocities observed from stationary inertial frame S attached to the road and moving inertial frame S' attached to the first car.

Consider a situation common in everyday life. Two cars travel along a road, both moving at a constant velocity. See Figure 1. At some particular moment, they are separated by 200 meters. The car in front is traveling at 22 meters per second and the car behind is traveling at 30 meters per second. If we want to find out how long it will take the second car to catch up with the first, there are three obvious "frames of reference" that we could choose.

First, we could observe the two cars from the side of the road. We define our "frame of reference" S as follows. We stand on the side of the road and start a stop-clock at the exact moment that the second car passes us, which happens to be when they are a distance d = 200 m apart. Since neither of the cars are accelerating, we can determine their positions by the following formulas, where $x 1 (t)$ is the position in meters of car one after time t seconds and $x 2 (t)$ is the position of car two after time t.

$x_1(t)= d + v_1 t = 200\ + \ 22t\ ; \quad x_2(t)= v_2 t = 30t$

Notice that these formulas predict at t = 0 s the first car is 200 m down the road and the second car is right beside us, as expected. We want to find the time at which $x 1 = x 2$ . Therefore we set $x 1 = x 2$ and solve for $t$ , that is:

$200 + 22 t = 30t \quad$

$8t = 200 \quad$

$t = 25 \quad \mathrm{seconds}$

Alternatively, we could choose a frame of reference S' situated in the first car. In this case, the first car is stationary and the second car is approaching from behind at a speed of v₂ − v₁ = 8 m / s. In order to catch up to the first car, it will take a time of d /( v₂ − v₁) = 200 / 8 s, that is, 25 seconds, as before. Note how much easier the problem becomes by choosing a suitable frame of reference. The third possible frame of reference would be attached to the second car. That example resembles the case just discussed, except the second car is stationary and the first car moves backward towards it at 8 m / s.

It would have been possible to choose a rotating, accelerating frame of reference, moving in a complicated manner, but this would have served to complicate the problem unnecessarily. It is also necessary to note that one is able to convert measurements made in one coordinate system to another. For example, suppose that your watch is running five minutes fast compared to the local standard time. If you know that this is the case, when somebody asks you what time it is, you are able to deduct five minutes from the time displayed on your watch in order to obtain the correct time. The measurements that an observer makes about a system depend therefore on the observer's frame of reference (you might say that the bus arrived at 5 past three, when in fact it arrived at three).

Remarks

See also: Special theory of relativity and General theory of relativity

It is important to note that there were a number of assumptions made about the various inertial frames of reference. Newton for instance believed in a concept known as universal time. This is best explained by an example. Suppose that you own two clocks, which both tick at exactly the same rate. You synchronise them so that they both display the exact same time. The two clocks are now separated and one clock is on a fast moving train, travelling at constant velocity towards the other. According to Newton, these two clocks will still tick at the same rate and will both show the same time. Newton says that the rate of time as measured in one frame of reference should be the same as the rate of time in another. That is, there exists a "universal" time and all other times in all other frames of reference will run at the same rate as this universal time irrespective of their position and velocity. This concept was later disproven by Einstein in his special theory of relativity (1905) where he developed transformations between inertial frames of reference based of their relative displacement and relative velocity (Lorentz transformations).

It is also important to note that the definition of inertial reference frame (defined as one in which a free particle travels in a straight line at constant speed) does not include a requirement that the inertial reference frame must exist in three dimensional Euclidean space. This was another of Newton's assumptions which would later be disproven. As an example of why this is important, let us consider the Non-Euclidean geometry of a sphere. In this geometry, two free particles may begin at the same point on the sphere, travelling with the same constant speed in different directions. After a length of time, the two particles will collide at the opposite side of the sphere. Both free particles were travelling with a constant velocity and no forces were acting. No acceleration occurred and so Newton's first law held true. This means that the particles were in inertial frames of reference. Since no forces were acting, it was the geometry of the situation which caused the two particles to meet each other again. In a similar way, it is now believed that we exist in a four dimensional geometry known as spacetime. It is believed that the curvature of this 4D space is responsible for the way in which two bodies with mass will meet together even if no forces are acting. In Newtonian mechanics, this is explained by a force known as gravity.

Additional examples

For a simple example, consider two people standing, facing each other on either side of a north-south street. A car drives past them heading south. For the person facing east, the car was moving toward the right. However, for the person facing west, the car was moving toward the left. This discrepancy is because the two people used two different frames of reference from which to investigate this system.

For a more complex example, consider Alfred, who is standing on the side of a road watching a car drive past him from left to right. In his frame of reference, Alfred defines the spot where he is standing as the origin, the road as the x-axis and the direction in front of him as the positive y-axis. To him, the car moves along the x axis with some velocity v in the positive x-direction. Alfred's frame of reference is considered an inertial frame of reference because he is not accelerating (ignoring effects such as Earth's rotation and gravity).

Now consider Betsy, the person driving the car. Betsy, in choosing her frame of reference, defines her location as the origin, the direction to her right as the positive x-axis, and the direction in front of her as the positive y-axis. In this frame of reference, it is Betsy who is stationary and the world around her that is moving - for instance, as she drives past Alfred, she observes him moving with velocity v in the negative y-direction. If she is driving north, then north is the positive y-direction; if she turns east, east becomes the positive y-direction.

Now assume Candace is driving her car in the opposite direction. As she passes by him, Alfred measures her acceleration and finds it to be a in the negative x-direction. Assuming Candace's acceleration is constant, what acceleration does Betsy measure? If Betsy's velocity v is constant, she is in an inertial frame of reference, and she will find the acceleration to be the same - in her frame of reference, a in the negative y-direction. However, if she is accelerating at rate A in the negative y-direction (in other words, slowing down), she will find Candace's acceleration to be a' = a - A in the negative y-direction - a smaller value than Alfred has measured. Similarly, if she is accelerating at rate A in the positive y-direction (speeding up), she will observe Candace's acceleration as a' = a + A in the negative y-direction - a larger value than Alfred's measurement.

Frames of reference are especially important in special relativity, because when a frame of reference is moving at some significant fraction of the speed of light, then the flow of time in that frame does not necessarily apply in another reference frame. The speed of light is considered to be the only true constant between moving frames of reference.

Non-inertial frames

See also: Fictitious force

Here we consider the relation between inertial and non-inertial observational frames of reference.

An accelerated frame of reference is often delineated as being the "primed" frame, and all variables that are dependent on that frame are notated with primes, e.g. x' , y' , a' .

The vector from the origin of an inertial reference frame to the origin of an accelerated reference frame is commonly notated as R. Given a point of interest that exists in both frames, the vector from the inertial origin to the point is called r, and the vector from the accelerated origin to the point is called r'. From the geometry of the situation, we get

$\mathbf r = \mathbf R + \mathbf r'$

Taking the first and second derivatives of this, we obtain

$\mathbf v = \mathbf V + \mathbf v'$

$\mathbf a = \mathbf A + \mathbf a'$

where V and A are the velocity and acceleration of the accelerated system with respect to the inertial system and v and a are the velocity and acceleration of the point of interest with respect to the inertial frame.

These equations allow transformations between the two coordinate systems; for example, we can now write Newton's second law as

$\mathbf F = m\mathbf a = m\mathbf A + m\mathbf a'$

When there is accelerated motion due to a force being exerted there is manifestation of inertia. If an electric car designed to recharge its battery system when decelerating is switched to braking, the batteries are recharged, illustrating the physical strength of manifestation of inertia. However, the manifestation of inertia does not prevent acceleration (or deceleration), for manifestation of inertia occurs in response to change in velocity due to a force. Seen from the perspective of a rotating frame of reference the manifestation of inertia appears to exert a force (either in centrifugal direction, or in tangential direction, the Coriolis effect). Actually the force exerted on the object that keeps the object's motion in sync with the rotating frame elicits manifestation of inertia. If there is insufficient force to keep the object's motion in sync with the rotating frame, then seen from the perspective of the rotating frame there is an apparent acceleration. Whenever manifestation of inertia appears to act as a force it is labeled as a fictitious force. Inertia is very much real, of course, but unlike force it never accelerates an object.

A common sort of accelerated reference frame is a frame that is both rotating and translating (an example is a frame of reference attached to a CD which is playing while the player is carried). This arrangement leads to the equation (see Fictitious force for a derivation):

$\mathbf a = \mathbf a' + \dot{\boldsymbol\omega} \times \mathbf r' + 2\boldsymbol\omega \times \mathbf v' + \boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r') + \mathbf A_0$

or, to solve for the acceleration in the accelerated frame,

$\mathbf a' = \mathbf a - \dot{\boldsymbol\omega} \times \mathbf r' - 2\boldsymbol\omega \times \mathbf v' - \boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r') - \mathbf A_0$

Multiplying through by the mass m gives

$\mathbf F' = \mathbf F_\mathrm{physical} + \mathbf F'_\mathrm{Euler} + \mathbf F'_\mathrm{Coriolis} + \mathbf F'_\mathrm{centripetal} - m\mathbf A_0$

where

$\mathbf F'_\mathrm{Euler} = -m\dot{\boldsymbol\omega} \times \mathbf r'$ (Euler force)

$\mathbf F'_\mathrm{Coriolis} = -2m\boldsymbol\omega \times \mathbf v'$ (Coriolis force)

$\mathbf F'_\mathrm{centrifugal} = -m\boldsymbol\omega \times (\boldsymbol\omega \times \mathbf r')=m(\omega^2 \mathbf r'- (\boldsymbol\omega \cdot \mathbf r')\boldsymbol\omega)$ (centrifugal force)

Particular frames of reference in common use

International Terrestrial Reference Frame
International Celestial Reference Frame
In fluid mechanics, Eulerian Reference Frame and Lagrangian Reference Frame

In fields other than Physics

Footnotes

^ In very general terms, a coordinate system is a set of arcs xⁱ = xⁱ (t) in a complex Lie group; see Lev Semenovich Pontri͡agin. L.S. Pontryagin: Selected Works Vol. 2: Topological Groups, 3rd Edition, Gordon and Breach, p. 429. ISBN 2881241336. . Less abstractly, a coordinate system in a space of n-dimensions is defined in terms of a basis set of vectors {e₁, e₂,… e_n}; see Edoardo Sernesi, J. Montaldi (1993). Linear Algebra: A Geometric Approach. CRC Press, p. 95. ISBN 0412406802. As such, the coordinate system is a mathematical construct, a language, that may be related to motion, but has no necessary connection to motion.
^ J X Zheng-Johansson and Per-Ivar Johansson (2006). Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces. Nova Publishers, p. 13. ISBN 1594542600.
^ Jean Salençon, Stephen Lyle (2001). Handbook of Continuum Mechanics: General Concepts, Thermoelasticity. Springer, p. 9. ISBN 3540414436.
^ Akhlesh Lakhtakia (1993). Essays on the Formal Aspects of Electromagnetic Theory. World Scientific, p.149. ISBN 9810208545.
^ Graham Nerlich (1994). What Spacetime Explains: Metaphysical essays on space and time. Cambridge University Press, p.64. ISBN 0521452619.
^ Katherine Brading & Elena Castellani (2003). Symmetries in Physics: Philosophical Reflections. Cambridge University Press, p. 417. ISBN 0521821371. }
^ William Barker & Roger Howe (2008). Continuous symmetry: from Euclid to Klein, p. 18 ff. ISBN 0821839004.
^ Arlan Ramsay & Robert D. Richtmyer (1995). Introduction to Hyperbolic Geometry. Springer, p. 11. ISBN 0387943390.
^ Shigeyuki Morita, Teruko Nagase, Katsumi Nomizu (2001). Geometry of Differential Forms. American Mathematical Society Bookstore, p. 12. ISBN 0821810456.
^ Granino Arthur Korn, Theresa M. Korn (2000). Mathematical handbook for scientists and engineers : definitions, theorems, and formulas for reference and review. Courier Dover Publications, p. 169. ISBN 0486411478.
^ See Encarta definition
^ Achilleus Papapetrou (1974). Lectures on General Relativity. Springer, p. 5. ISBN 9027705402.
^ Wilford Zdunkowski & Andreas Bott (2003). Dynamics of the Atmosphere. Cambridge University Press, p. 84. ISBN 052100666X.
^ A. I. Borisenko, I. E. Tarapov, Richard A. Silverman (1979). Vector and Tensor Analysis with Applications. Courier Dover Publications, p. 86. ISBN 0486638332.
^ See A. Kumar, Shrish Barve (2003). How and why in Basic Mechanics. Orient Longman, p. 115. ISBN 8173714207.
^ Chris Doran & Anthony Lasenby (2003). Geometric Algebra for Physicists. Cambridge University Press, §5.2.2, p. 133. ISBN 978-0-521-71595-9. .
^ For example, Møller states: "Instead of Cartesian coordinates we can obviously just as well employ general curvilinear coordinates for the fixation of points in physical space.…we shall now introduce general "curvilinear" coordinates xⁱ in four-space…." C. Møller (1952). The Theory of Relativity. Oxford University Press, p. 222 and p. 233.
^ Alan P. Lightman, R. H. Price & William H. Press (1975). Problem Book in Relativity and Gravitation. Princeton University Press, p. 15. ISBN 069108162X.
^ Richard L Faber (1983). Differential Geometry and Relativity Theory: an introduction. CRC Press, p. 211. ISBN 082471749X.

Distortions can vary from place to place, with gravity appearing to be the common cause. In fact, General relativity predicts a frame-dragging effect (aka Lense-Thirring effect).^clarify

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