Frame of Reference

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

Different aspects of "frame of reference" Coordinate system attached as a modifier Cartesian Frame of Reference State of motion Rotating frame of reference

Different aspects of "frame of reference" Transformation of the frames Galilean frame of reference Scales Macroscopic and Microscopic frames of reference

Observational equipment Three Concepts Observational frames of reference Coordinate system Observational equipment

Observational frame of reference An observational frame (such as an inertial frame or non-inertial frame of reference) is a physical concept related to state of motion. State of motion Observational frame of reference Inertial Non- Inertial

Coordinate System A coordinate system is a mathematical concept, amounting to a choice of language used to describe observations. Consequently, an observer in an observational frame of reference can choose to employ any coordinate system (Cartesian, polar, curvilinear, generalized, …) to describe observations made 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. This viewpoint can be found elsewhere as well. Which is not to dispute that some coordinate systems may be a better choice for some observations than are others.

Observed, Observational Apparatus, and observer's state of motion Choice of what to measure and with what observational apparatus is a matter separate from the observer's state of motion and choice of coordinate system.

Reference Frame Jean Salençon, Stephen Lyle (2001). Handbook of Continuum Mechanics: General Concepts, Thermoelasticity. Springer. p. 9.

Inertial frame of reference 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. These relativistic and Newtonian transformations are expressed in spaces of general dimension in terms of representations of the Poincaré group and of the Galilean group.

Inertial frame of reference In physics, an inertial frame of reference is a member of the subset of reference frames with the property that every physical law takes the same form in each such frame. The measurements that an observer makes about a system dose not depend therefore on the observer's inertial frame of reference.

Example of Inertial Frame of Reference 30 m/sec 22 m/sec 200 m t=25 s No matter where to situate the frame of reference. In 25 seconds the second car catches up with the first.

Example of Non- Inertial Frame of Reference

Another example: CD which is playing while the player is carried z´ x y z r r´ R

Fictitious Forces These terms all have these properties: they vanish when ω = 0; that is, they are zero for an inertial frame (which, of course, does not rotate); they take on a different magnitude and direction in every rotating frame, depending upon its particular value of ω; they are ubiquitous in the rotating frame (affect every particle, regardless of circumstance); and they have no apparent source in identifiable physical sources, in particular, matter. Also, fictitious forces do not drop off with distance (unlike, for example, nuclear forces or electrical forces). For example, the centrifugal force that appears to emanate from the axis of rotation in a rotating frame increases with distance from the axis.

Fictitious vs. Real Forces All observers agree on the real forces, F; only non-inertial observers need fictitious forces. The laws of physics in the inertial frame are simpler because unnecessary forces are not present.

Some Remarks Newton The Universal Time Generalized by Einstein Euclidean Space 4D Space (Einstein)

Newton’s Reference Frame Fixed stars (at rest relative to absolute space) Newton’s frame of reference So Newton’s laws of motion hold

Newton’s Reference Frame In contrast, in frames accelerating with respect to the fixed stars, an important case being frames rotating relative to the fixed stars, the laws of motion did not hold in their simplest form, but had to be supplemented by the addition of fictitious forces, for example, the Coriolis force and the centrifugal force.

Fixed stars are not fixed Those in Milky Way which turn with the galaxy Those outside the galaxy Participating in expansion of the Universe and peculiar velocities

New Approach to the Inertial Reference Frame The concept of inertial frames of reference is no longer tied to either the fixed stars or to absolute space. Rather, The identification of an inertial frame is based upon the simplicity of the laws of physics in the frame. In particular, the absence of fictitious forces is their identifying property.

Inertial frames in special relativity and in Newtonian mechanics A brief comparison SPECIAL RELATIVITY Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.

Inertial frames in special relativity and in Newtonian mechanics A brief comparison NEWTONIAN MECHANICS The principle of simplicity can be used within Newtonian physics as well as in special relativity: The laws of Newtonian mechanics do not always hold in their simplest form...If, for instance, an observer is placed on a disc rotating relative to the earth, he/she will sense a 'force' pushing him/her toward the periphery of the disc, which is not caused by any interaction with other bodies. Here, the acceleration is not the consequence of the usual force, but of the so-called inertial force. Newton's laws hold in their simplest form only in a family of reference frames, called inertial frames. This fact represents the essence of the Galilean principle of relativity:    The laws of mechanics have the same form in all inertial frames. Ernest Nagel (1979). The Structure of Science. Hackett Publishing. p. 212.

In practical terms, the equivalence of inertial reference frames means that scientists within a box moving uniformly cannot determine their absolute velocity by any experiment (otherwise the differences would set up an absolute standard reference frame). According to this definition, supplemented with the constancy of the speed of light, inertial frames of reference transform among themselves according to the Poincaré group of symmetry transformations, of which the Lorentz transformations are a subgroup. In Newtonian mechanics, which can be viewed as a limiting case of special relativity in which the speed of light is infinite, inertial frames of reference are related by the Galilean group of symmetries.

ABSOLUTE TIME AND SPACE Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time … (Newton, in Philosophiae Naturalis Principia Mathematica)

ABSOLUTE TIME AND SPACE Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies: and which is vulgarly taken for immovable space … (Newton, in Philosophiae Naturalis Principia Mathematica)

ABSOLUTE TIME AND SPACE Absolute space and time do not depend upon physical events Absolute space and time do not depend observer’s state of motion

ABSOLUTE TIME AND SPACE: Objections 1. The existence of absolute space contradicts the internal logic of classical mechanics since, according to Galilean principle of relativity, none of the inertial frames can be singled out.

ABSOLUTE TIME AND SPACE: Objections 2. Absolute space does not explain inertial forces since they are related to acceleration with respect to any one of the inertial frames.

ABSOLUTE TIME AND SPACE: Objections 3. Absolute space acts on physical objects by inducing their resistance to acceleration but it cannot be acted upon.

Special and General Theories of Relativity Special relativity theory connected the two and showed both to be dependent upon the observer's state of motion. In Einstein's theories, the ideas of absolute time and space were superseded by the notion of spacetime in special relativity, and by dynamically curved spacetime in general relativity.

Classical Mechanics Very short review The kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its current velocity. In classical mechanics, the kinetic energy of a point object (an object so small that its mass can be assumed to exist at one point), or a non-rotating rigid body, is given by the equation

Classical Mechanics Very short review The kinetic energy of an object is related to its momentum by the equation:

Conservation of energy Conservation of linear momentum Conservation laws Exact Conservation of energy Conservation of linear momentum Conservation of angular momentum Conservation of electric charge Conservation of color charge Conservation of weak isospin Conservation of probability

Conservation of baryon number Conservation of lepton number Conservation laws Approximate Conservation of mass Conservation of baryon number Conservation of lepton number Conservation of flavor Conservation of parity CP symmetry

Conservation of energy The law of conservation of energy states that the total amount of energy in a closed system remains constant. A consequence of this law is that energy cannot be created nor destroyed. The only thing that can happen with energy in a closed system is that it can change form, for instance kinetic energy can become thermal energy.

Conservation of linear momentum The law of conservation of linear momentum is a fundamental law of nature, and it states that the total momentum of a closed system of objects (which has no interactions with external agents) is constant. One of the consequences of this is that the center of mass of any system of objects will always continue with the same velocity unless acted on by a force from outside the system.

Conservative force and potential energy F= - dU/dx E= K+U

Angular Momentum x z y p r L=r P

In a closed system angular momentum is constant. Conservation of angular momentum In a closed system angular momentum is constant.