Proper frame

Last updated February 06, 2024

A proper frame, or comoving frame, is a frame of reference that is attached to an object. The object in this frame is stationary within the frame, which is useful for many types of calculations.

For example, a freely falling elevator is a proper frame for a free-falling object in the elevator, while the surface of the Earth is not. But, for an object on the Earth's surface, the Earth's surface is a proper frame while the falling elevator is not a proper frame. Proper frames can be inertial and non-inertial, as in the example above.

The use of a proper frame is essential for the investigation of physical laws within the framework of general relativity.

The term comoving frame is also a good description of a non-inertial frame, which is useful for many of the same uses as we mentioned previously. One advantage of proper frame and comoving frame is that the two frames must always maintain the same spatial position (i. "in the frame" - e.g. on the same frame of reference). This includes that the frame must always be in position in the spacetime frame and thus the spacetime can be viewed as having "no axis". As our first example of a proper frame, one uses the following frame to find the Earth:

The Earth is situated in the center with respect to the observer (or our point of reference) of our next example, the Sun is at the bottom.

𝜕 is described as the set of sets that have the property that the motion vectors of an object are conserved. 𝜕 can be thought of as the set of sets (including proper frames) of all possible motions of a given object, such that a proper frame always results.^[1]

In quantum field theory and many fields of physics, such as electromagnetism, it is often referred to as the "comoving frame" of a particle. 𝜕 can be thought of as the unique set of frames that are conserved under gravity, allowing that the particles of gravitation do not collapse on an object after the initial contact (for example, they remain in the frame they have been suspended in).^[2]^[1]

An "inertial frame" has an inertial reference vector to a fixed point in the spacetime continuum. For example, suppose I place an object on a horizontal line and extend the line upwards. The line originates at an point x at the center of vertical symmetry in the plane perpendicular to the horizontal plane (and the line continues downwards to the bottom of the vertical line) at x = −X where x is the horizontal line velocity on my line.

Then if the object is placed on horizontal line X a new object (with an inertial reference vector perpendicular to the horizontal line) that originates as if it were placed on the horizontal line X would be brought to a line point A at x = −A − x. This would produce a new object that originates vertically from an empty point or point A at point A, i.e. a new object that has a higher momentum than the one that existed at point A . This principle holds whether the point A is horizontal line X, a fixed point such as X at right angles to a line from this plane or any other fixed point, such as the bottom plane of a plane or some part of spacetime.^[3]

Consider what this means; if I place the object at x = +V there exists a vector of velocities in the plane parallel to that line; I add a vector to the vertical line that points in that direction; and then I continue moving down the same line and point my object on that horizontal line a distance T?

This principle holds whether a fixed point is horizontal line X at right angles to a fixed point at a point such as X at right angles with the plane of a horizontal plane. A fixed point would be placed on X using any means suitable for horizontal line X, such as applying a line to the end point of one object that contains an inertial reference vector along that line, applying a line to the end of one object that contains an inertial reference vector along this line on the right side of the plane parallel to the plane, using a line to the centerline or center of a plane, or a line to any other straight straight horizontal line.^[4]

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In classical physics and special relativity, an inertial frame of reference is a frame of reference not undergoing any acceleration. It is a frame in which an isolated physical object—an object with zero net force acting on it—is perceived to move with a constant velocity or, equivalently, it is a frame of reference in which Newton's first law of motion holds. All inertial frames are in a state of constant, rectilinear motion with respect to one another; in other words, an accelerometer moving with any of them would detect zero acceleration. It has been observed that celestial objects which are far away from other objects and which are in uniform motion with respect to the cosmic microwave background radiation maintain such uniform motion.

In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 treatment, the theory is presented as being based on just two postulates:

The laws of physics are invariant (identical) in all inertial frames of reference.
The speed of light in vacuum is the same for all observers, regardless of the motion of light source or observer.

In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects such as how different observers perceive where and when events occur.

<span class="mw-page-title-main">Rotation</span> Movement of an object around an axis

Rotation or rotational motion is the circular movement of an object around a central line, known as axis of rotation. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a center of rotation. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation, in contrast to rotation around a fixed axis.

In physics and astronomy, a frame of reference is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points―geometric points whose position is identified both mathematically and physically.

In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference.

In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects. Comoving distance factors out the expansion of the universe, giving a distance that does not change in time due to the expansion of space. Proper distance roughly corresponds to where a distant object would be at a specific moment of cosmological time, which can change over time due to the expansion of the universe. Comoving distance and proper distance are defined to be equal at the present time. At other times, the Universe's expansion results in the proper distance changing, while the comoving distance remains constant.

The world line of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept of modern physics, and particularly theoretical physics.

Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGerald contraction and is usually only noticeable at a substantial fraction of the speed of light. Length contraction is only in the direction in which the body is travelling. For standard objects, this effect is negligible at everyday speeds, and can be ignored for all regular purposes, only becoming significant as the object approaches the speed of light relative to the observer.

<span class="mw-page-title-main">Absolute space and time</span> Theoretical foundation of Newtonian mechanics

Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred frame.

A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame. Fictitious forces are invoked to maintain the validity and thus use of Newton's second law of motion, in frames of reference which are not inertial.

The ladder paradox is a thought experiment in special relativity. It involves a ladder, parallel to the ground, travelling horizontally at relativistic speed and therefore undergoing a Lorentz length contraction. The ladder is imagined passing through the open front and rear doors of a garage or barn which is shorter than its rest length, so if the ladder was not moving it would not be able to fit inside. To a stationary observer, due to the contraction, the moving ladder is able to fit entirely inside the building as it passes through. On the other hand, from the point of view of an observer moving with the ladder, the ladder will not be contracted, and it is the building which will be Lorentz contracted to an even smaller length. Therefore, the ladder will not be able to fit inside the building as it passes through. This poses an apparent discrepancy between the realities of both observers.

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A non-inertial reference frame is a frame of reference that undergoes acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will, in general, detect a non-zero acceleration. While the laws of motion are the same in all inertial frames, in non-inertial frames, they vary from frame to frame depending on the acceleration.

In geometry, the orientation, attitude, bearing, direction, or angular position of an object – such as a line, plane or rigid body – is part of the description of how it is placed in the space it occupies. More specifically, it refers to the imaginary rotation that is needed to move the object from a reference placement to its current placement. A rotation may not be enough to reach the current placement, in which case it may be necessary to add an imaginary translation to change the object's position. The position and orientation together fully describe how the object is placed in space. The above-mentioned imaginary rotation and translation may be thought to occur in any order, as the orientation of an object does not change when it translates, and its position does not change when it rotates.

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<span class="mw-page-title-main">Proper velocity</span> Ratio in relativity

In relativity, proper velocityw of an object relative to an observer is the ratio between observer-measured displacement vector $and proper time τ elapsed on the clocks of the traveling object:$

Earth-centered inertial (ECI) coordinate frames have their origins at the center of mass of Earth and are fixed with respect to the stars. "I" in "ECI" stands for inertial, in contrast to the "Earth-centered – Earth-fixed" (ECEF) frames, which remains fixed with respect to Earth's surface in its rotation, and then rotates with respect to stars.

In special relativity, an observer is a frame of reference from which a set of objects or events are being measured. Usually this is an inertial reference frame or "inertial observer". Less often an observer may be an arbitrary non-inertial reference frame such as a Rindler frame which may be called an "accelerating observer".

References

1 2 Patrick Cornille (1993). "Inhomogenous waves and Maxwell's equations". In Akhlesh Lakhtakia (ed.). Essays on the Formal Aspects of Electromagnetic Theory. World Scientific. p. 149. ISBN 981-02-0854-5.
↑ Comoving frames and the Lorentz–Fitzgerald contraction American Journal of Physics 87, 5 (2019); https://doi.org/10.1119/1.5082535
↑ Rudman, John W. (1999), The General Relativity of General Relativity, Princeton: Princeton University Press
↑ Meadow, Daniel A., and J. S. Huxley (1982), 'Introduction to Einstein's Theory of Relativity', In: J. S. Huxley (ed.), Relativity Theory, London: Chapman & Hall, ISBN 0-415-0288-9

Proper frame

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References

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