Frame of reference

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In physics, the reference frame (or reference frame ) consists of an abstract coordinate system and a set of physical reference points that uniquely fix (locate and direct) coordinate systems and standardize measurements.

In n dimensions, the reference point n 1 is sufficient to fully specify the frame of reference. Using the Cartesian coordinate, the reference frame can be defined by reference point at the origin and reference points on a single unit of distance along each of the coordinate axes n.

In Einsteinian relativity, a frame of reference is used to determine the relationship between a moving observer and a phenomenon or phenomenon under observation. In this context, this phrase is often the "observational reference frame " (or " observational reference frame "), which implies that the observer is resting in a frame, though not necessarily in its original place. The relativistic reference frames include (or implies) coordinate time, which does not correspond to the relative frames that move relative to each other. This situation is different from the relativity of Galilee, where all the possible coordination times are essentially equivalent.

Video Frame of reference

Different aspects of "reference frame"

The need to differentiate between the various meanings of the "terms of reference" has led to various terms. For example, sometimes the type of coordinate system is attached as a modifier, as in the Cartesian reference frame . Sometimes the state of motion is emphasized, as in rotating the terms of reference . Sometimes the way it turns into a frame that is considered to be emphasized is emphasized as in the Galilean reference framework . Sometimes frames are distinguished based on their observation scale, as in macroscopic and microscopic reference frames .

In this article, the term observational reference frame is used when the emphasis is on the motion state rather than on the coordinate or observational character or observational apparatus. In this sense, the observational reference frame allows the study of the effect of motion on the entire family of coordinate systems that can be attached to this frame. On the other hand, the coordinate system can be used for many purposes where motion is not a major concern. For example, coordinate systems can be adopted to take advantage of the symmetry of a system. In a broader perspective, the formulation of many problems in physics uses general coordinates, normal mode, or eigenvectors, which are only indirectly related to space and time. It seems useful to divorce various aspects of the terms of reference for the discussion below. We therefore take the observational reference frame, coordinate system, and observation tools as independent concepts, separated as below:

• Observational frames (such as inertial frames or non-inertial reference frames) are physical concepts related to the state of motion.
• The coordinate system is a mathematical concept, which is the choice of language used to describe observation. Consequently, observers within an observational reference frame may choose to use any coordinate system (Cartesian, polar, curved, general,...) to illustrate the observations made from that frame of reference. This change in the choice of coordinate system does not alter the observer's status of motion, and therefore does not require any change in the observational reference frame observation . This viewpoint can be found elsewhere as well. That is not to argue that some coordinate systems might be a better choice for some observations than others.

• The choice of what is measured and with the observational apparatus is a matter separate from the observer's status of motion and the choice of coordinate system.

Berikut adalah kutipan yang berlaku untuk menggerakkan bingkai pengamatan ${\ displaystyle {\ mathfrak {R}}}  ÂÂ$ dan berbagai sistem koordinat tiga ruang Euclidean yang terkait [ R , R? , dll. ]:

dan ini pada utilitas memisahkan pengertian ${\ displaystyle {\ mathfrak {R}}}  ÂÂ$ dan [ R , R? , dll. ]:

dan ini, juga pada perbedaan antara ${\ displaystyle {\ mathfrak {R}}}  ÂÂ$ dan [ R , R? , dll. ]:

and from J. D. Norton:

This discussion was taken outside a simple space-time coordinate system by Brading and Castellani. The expansion to coordinate systems using common coordinates underlies the Hamiltonian and Lagrangian formulations of quantum field theory, classical relativistic mechanics, and quantum gravity.

Coordinate system

Although the term "coordinate system" is often used (mainly by physicists) in a non-technical sense, the term "coordinate system" does have precise meanings in mathematics, and sometimes it is what physicists also mean.

Sistem koordinat dalam matematika adalah segi geometri atau aljabar, khususnya, properti manifold (misalnya, dalam fisika, ruang konfigurasi atau ruang fase). Koordinat titik r dalam ruang n -dimensi hanyalah serangkaian himpunan bilangan n :

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

In the common Banach space, these numbers can be (for example) coefficients in functional expansions such as the Fourier series. In physical problems, they can be the spatial coordinates or normal mode amplitudes. In robotic design, they can be relative rotational angles, linear displacements, or joint deformations. Here we will assume this coordinate can be associated with a Cartesian coordinate system by a set of functions:

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

di mana x , y , z , dll. adalah n koordinat Cartesian dari inti nya. Dengan fungsi ini, koordinat permukaan ditentukan oleh relasi:

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

which can be normalized into units of length. For more details, see the curved coordinates.

Coordinate surfaces, coordinate lines, and base vectors are components of the coordinate system . If the basis vector is orthogonal at any point, the coordinate system is an orthogonal coordinate system.

Aspek penting dari sistem koordinat adalah tensor metrik g _ik , yang menentukan panjang busur ds dalam sistem koordinat dalam hal koordinatnya:

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

where the recurring index is added up.

As can be seen from this statement, the coordinate system is a mathematical construct, a part of an axiomatic system. No connection is required between coordinate system and physical movement (or any other aspect of reality). However, the coordinate system can include time as a coordinate, and can be used to describe motion. Thus, the Lorentz transformation and the Galilean transformation can be seen as coordinate transformations.

General and specific topics of the coordinate system can be pursued following the See also link below.

Observation reference frame

An observational reference frame , often referred to as a physical reference frame , a reference frame , or just a frame , is a physical concept associated with the observer and observer status. Here we adopt the view expressed by Kumar and Barve: the reference frame of observation is characterized only by the state of motion . However, there is a lack of sound at this point. In special relativity, differences are sometimes made between observers and frames . According to this view, a frame is an observer of plus a coordinate lattice constructed to be an orthonormal orthogonal vertex vector perpendicular to a time-like vector. View Doran. This limited view is not used here, and is not universally adopted even in the discussion of relativity. In general relativity the use of common coordinate systems is common (see, for example, Schwarzschild solutions for gravitational fields outside isolated spheres).

There are two types of terms of reference: inertia and non-inertia. The inertial reference framework is defined as one in which all the laws of physics take the simplest form. In special relativity, this framework is related to the Lorentz transformation, which dis parametri by speed. In Newtonian mechanics, a more limited definition simply requires that Newton's first law apply; ie, the inertial framework of Newton is one in which the free particles move in a straight line at a constant velocity, or are resting. These frames are related to the Galilean transformation. This relativist and Newtonian transformation is expressed in the general dimensional space in terms of representations of the PoincarÃÆ'Â and the Galilean groups.

In contrast to the inertial framework, a non-inertial reference frame is a framework in which fictitious forces should be used to explain observations. An example is an observation frame centered at the point on the surface of the Earth. This frame of reference orbits around the center of the Earth, which introduces a fictitious force known as Coriolis force, centrifugal force, and gravitational force. (All these forces including gravity disappear in a truly inertial reference frame, which is one of the free fall.)

Measurement tools

The next aspect of the frame of reference is the role of measuring instruments (eg, clocks and bars) attached to the frame (see Norton quote above). This question is not addressed in this article, and is of particular interest in quantum mechanics, where the relationship between observer and measurement is still under discussion (see measurement problem).

In physics experiments, the frame of reference in which a laboratory measurement device is resting is usually referred to as a laboratory framework or simply a "lab skeleton." An example would be a frame where the detector for the particle accelerator is resting. The laboratory frame in some experiments is an inertial skeleton, but not necessary (eg a laboratory on Earth's surface in many non-inertial physics experiments). In particle physics experiments, it is often useful to alter the energy and particle moments of the lab frame in which they are measured, to the center of the "COM frame" momentum frame where calculations are sometimes simplified, since potentially all kinetic energy still present in the COM frame can be used for create new particles.

In this connection it can be noted that clocks and trunks are often used to describe gauges in mind, in practice replaced by a much more complicated and indirect metrology connected to vacuum properties, and using atomic clocks that operate according to standard models and which must corrected for gravity time dilation. (See seconds, meter and kilogram).

In fact, Einstein feels that clocks and sticks are only a wise measure and they must be replaced by more basic entities based on, for example, atoms and molecules.

Maps Frame of reference

Type

• The reference frame remains
• Fixed space reference template
• Inertial reference frame
• Non-Inertial reference framework

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Example of inertial reference template

Simple example

Consider a common situation in everyday life. Two cars walked along the road, both moving at constant speed. See Figure 1. At some point, they are separated by 200 meters. The car in front was traveling at 22 meters per second and the car at the back was traveling at 30 meters per second. If we want to know how long it takes a second car to pursue the first one, there are three "terms of reference" we can choose from.

First, we can observe two cars from the side of the road. We define our "terms of reference" S as follows. We stood on the side of the road and started the stop-clock at the exact moment when the second car passed us, which happened when they were d = 200m apart. Since neither of the two cars accelerates, we can determine their position by the following formula, where $Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{          x              Â 1                         (          t        )           {\ displaystyle x_ {1} (t)}  Â} is the position in the car meter one by one t in seconds and _{          x               Â 2                         (          t        )               {\ displaystyle x_ {2} (t)}} is the position of the two cars after the t time.

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{          x              Â 1                         (          t        )         =          d                    Â     v              Â 1                           t         =         200         Ã,                 Ã,         22          t         Ã,         ;                          x               Â 2                         (          t        )         =            Â     v               Â 2                           t         =         30          t               {\ displaystyle x_ {1} (t) = d v_ {1} t = 200 \ \ 22t \; \ quad x_ {2} (t) = v_ {2} t = 30t}  Â}

Perhatikan bahwa rumus ini memprediksi pada t = 0 s mobil pertama berjarak 200 m di jalan dan mobil kedua tepat di samping kita, seperti yang diharapkan. Kami ingin to mention waktu di mana ${\ displaystyle x_ {1} = x_ {2}} Â Â$ . Oleh karena itu, kami menetapkan ${\ displaystyle x_ {1} = x_ {2}} Â$ dan pecahkan untuk ${\ displaystyle t} Â$ , yaitu:

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â200Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â22Â\; Â\; Â\; Â\; Â\; Â\; Ât\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â30Â\; Â\; Â\; Â\; Â\; Â\; Ât\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â}Â\; Â\; ÂÂ\; Â\; Â\{\backslash \; displaystyle\; 200\; 22t\; =\; 30t\; \backslash \; quad\}\; Â\; Â$

$Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â8Â\; Â\; Â\; Â\; Â\; Â\; Ât\; Â\; Â\; Â\; Â\; Â\; Â\; Â=Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â200Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â}Â\; Â\; ÂÂ\; Â\; Â\{\backslash \; displaystyle\; 8t\; =\; 200\; \backslash \; quad\}\; Â\; Â$

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

Or, we can choose the reference framework S? located in the first car. In this case, the first car of the stationary and the second car approaching from behind at the speed of v ₂ - v ₁ = 8 m/s . To pursue the first car, it will take time d / v ₂ - > 8 s , that is, 25 seconds, as before. Notice how easy the problem becomes by choosing a suitable frame of reference. A third reference frame that might be attached to the second car. The example resembles the case just discussed, except the second stationary car and the first car backing toward it at 8 m/s.

It will be possible to choose a frame of reference that spins and accelerates, moving in a complicated way, but this will help complicate unnecessary problems. It should also be noted that one can convert measurements made in one coordinate system to another coordinate system. For example, suppose that your clock runs five minutes faster than the local standard time. If you know that this is the case, when someone asks you what time, you can deduct five minutes from the time displayed on your watch to get the right time. An observer's measurement of a system depends on the terms of reference of the observer (you may say that the bus arrived at 5:30, when it actually arrived at three o'clock).

Additional examples

For a simple example that involves only the orientation of two observers, consider two people standing, facing each other on both sides of the north-south road. See Figure 2. A car drove past them heading south. For the person facing east, the car moves to the right. However, for the person facing west, the car moves to the left. This difference is because two people use two different reference frames to investigate this system.

For more complex examples involving observers in relative motion, consider Alfred, who stands by the side of the road seeing cars passing from left to right. In his terms of reference, Alfred defines the point at which he stands as the origin, the path as the x-axis and the direction in front of it as the positive y-axis. To him, the car moves along the axis x with some velocity v in the positive x direction. Alfred's reference frame is considered an inertial frame of reference because it does not accelerate (ignoring effects such as Earth's rotation and gravity).

Now watch Betsy, the guy driving the car. Betsy, in choosing her reference frame, defines her location as the origin, the direction to the right as positive x -axis, and the direction in front of it as positive y -axis. In this frame of reference, the silent Betsy and the moving world around him - for example, as he passes Alfred, he observes that he moves at v in negative y -direction. If he is driving north, then the north direction is positive y ; if he turns east, east becomes positive y .

Finally, for example a non-inertia observer, considers Candace to speed up his car. As he passed it, Alfred measured his acceleration and found it to be a in the negative x direction. Assuming the acceleration of Candace is constant, what acceleration does Betsy measure? If Betsy's velocity v is constant, he is in the frame of inertial reference, and he will find the acceleration to be the same as Alfred in his terms of reference, a in y -direction is negative. However, if he accelerates at a A level in a negative y direction (in other words, slows down), he will find Candace acceleration to be a? = a - A in the negative y direction - values ​​smaller than Alfred measure. Similarly, if he accelerates at the A level in the positive y direction (speeding), he will observe Candace acceleration as a? = a A in the negative y direction - a value larger than the size of Alfred.

The frame of reference is essential in special relativity, because when the frame of reference moves on some significant part of the speed of light, the time flow in the frame does not always apply in another frame. The speed of light is considered to be the only true constant between the mobile reference frame.

Comment

It is important to note some of the assumptions made above about various inertial reference terms. Newton, for example, uses universal time, as explained by the following example. Suppose you have two hours, both of which give the exact same number. You sync them so they display exactly the same time. The two clocks are now separated and one hour on a fast-moving train, walking at constant speed in the other direction. According to Newton, these two clocks will still tick at the same rate and both will show the same time. Newton says that the time rate measured in a single frame of reference must be equal to the rate of time elsewhere. That is, there is a "universal" time and all other times in all other terms of reference will run at the same level as this universal time regardless of position and speed. This concept of time and simultaneity was then generalized by Einstein in his special theory of relativity (1905) in which he developed a transformation between an inertial reference frame based on the universal nature of his physical law and economic expression (Lorentz transformation).

It is also important to note that the definition of an inertial reference frame can be extended beyond the three-dimensional Euclidean space. Newton assumes Euclidean space, but general relativity uses a more general geometry. As an example of why this is important, let's take a look at the ellipsoid geometry. In this geometry, "free" particles are defined as one at rest or traveling at a constant speed on a geodesic path. Two free particles can start at the same point on the surface, moving at the same constant speed in different directions. After some time, the two particles collide on the opposite side of the ellipsoid. Both "free" particles travel at constant speed, meeting the definition that no force acts. No acceleration occurred and therefore Newton's first law proved to be true. This means that the particles are within the framework of inertial reference. Since no force acts, it is the geometry of the situation that causes the two particles to meet again. In the same way, it is now common to illustrate that we exist in a four dimensional geometry known as spacetime. In this figure, the curvature of 4D space is responsible for the way in which two bodies with mass are drawn together even if no force works. This spacetime curvature replaces a force known as gravity in Newtonian mechanics and special relativity.

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Non-inertial frame

Here the relationship between inertial and non-inertial observation terms of reference is considered. The fundamental difference between these frames is the need in non-inertial frames for fictitious forces, as described below.

Accelerated reference frames are often described as "ready-made" frames, and all the variables that depend on the frame are denoted by primes, e.g. x? , y? , a? .

The vector from the origin of the inertial reference frame to the origin of the accelerated reference frame is usually denoted as R . Given the interesting point that exists in both frames, the vector from the origin of inertia to the point is called r , and the vector from origin that is accelerated to a point is called r? . From the geometry of the situation, we get

${\ displaystyle \ mathbf {r} = \ mathbf {R} \ mathbf {r} '}$

Mengambil turunan pertama dan kedua ini sehubungan dengan waktu, kita dapatkan

${\ displaystyle \ mathbf {v} = \ mathbf {V} \ mathbf {v} '} Â Â$

${\ displaystyle \ mathbf {a} = \ mathbf {A} \ mathbf {a} '} Â Â$

where V and A are speed and accelerated system acceleration with respect to the inertia system and v and a is the velocity and the acceleration of the point of concern with respect to the inertial framework.

Persamaan ini memungkinkan transformasi antara second sistem koordinat; misalnya, kita sekarang dapat menú hukum Newton kedua sebagai

${\ displaystyle \ mathbf {F} = m \ mathbf {a} = m \ mathbf {A} m \ mathbf {a} '} Â Â$

When there is an accelerated movement due to the given power there are manifestations of inertia. If an electric car designed to recharge its battery system as it slows down is transferred to braking, the battery is recharged, illustrating the physical strength of inertial manifestation. However, inertial manifestations do not prevent acceleration (or deceleration), for inertial manifestations occur in response to speed changes due to strength. Seen from the perspective of a rotating reference frame, the manifestation of inertia seems to exert force (either in the centrifugal direction, or in the orthogonal direction of object movement, the Coriolis effect).

A common type of accelerated frame of reference is a rotating and translating frame (for example, the reference frame attached to the CD being played when the player is taken). This setting leads to the equation (see fictitious power for an instance):

${\ displaystyle \ mathbf {a} = \ mathbf {a} '{\ dot {\ boldsymbol {\ omega}}} \ times \ mathbf {r} '2 {\ boldsymbol {\ omega}} \ times \ mathbf {v}' {\ boldsymbol {\ omega}} \ times {{\ boldsymbol {\ omega}} \ times \ mathbf {r} ') \ mathbf {A } _ {0}}$