Space-Time Symmetry

Properties of Space Three Dimensionality: If we consider any arbitrary point in space then maximum three perpendicular lines can be drawn from this point. These mutually perpendicular lines are called three axes of the coordinate system. Therefore position of an arbitrary point in space can be defined using three coordinates (x, y, z). So space is three dimensional. Flatness: Space is Flat. This means (i) In a right angle triangle, (Hypotenuse)2 = (Base)2 + (Normal)2 (ii) The sum of three angles of a triangle is equal to π radians. (iii) The shortest distance between two points in the space is a straight line. For most of Classical Mechanics Problems, the space is assumed to be completely flat. Isotropy: Isotropy of space means uniformity of direction. All the directions of space are equally preferred. E.g.: if we have a source of light in space, then it will send light rays in all the directions with equal speed.

Properties of Space Homogeneity: Homogeneity of space means space is alike everywhere. If an experiment is performed in space at one place then the identical experiment performed anywhere in space will give the identical results. Also If F=ma is valid in one coordinate system then it will also be valid in another coordinate system. Space Reflection: Space reflection means transformation of coordinates under which coordinates change sign. For example if in right handed coordinate system then in left handed coordinate system However laws of physics (e.g. F=ma) remains invariant.

Properties of Time One Dimensionality Time is specified by only one coordinate ‘t’. Hence time is one dimensional. Homogeneity (i) The laws of physics does not change with time. For example: If we perform an experiment today and repeat it after one month, the result of the experiment will be the same. (ii) interval of time are not affected by origin of time. One minute of today will be exactly equal to one minute of yesterday or tomorrow. Isotropy The laws of physics remains invariant by changing t to –t.

Conclusion The properties of space do not change with time. Time interval has same value for all times i.e., time flows uniformly. Free space is homogenous. Free space is isotropic. Free space has property of reflection. These symmetry properties of space & time are called space-time invariance principle. According to these, laws of nature are same at all points in space & for all times.

Taylor’s Theorem

Frame of Reference This train platform serves as the “rest” frame for Observers A and B, but it moves with a speed v towards Observer C, who stands on the roadside. A B C v

Homogeneity of Space & Newton’s Third law of motion Consider two particles A & B interacting with each other. Let be the position vectors of the particles in coordinate system S. The P.E. of interaction of two particles will be Consider another coordinate system S’ which is displaced infinitesimally through a displacement dr. The position vectors of the particles in S’ will be Then P.E. of interaction of two particles will be

Z’ S’ Z S Y’ A O’ O Y X’ B X

Homogeneity of Space & Newton’s Third law of motion According to Principle of Homogeneity

Homogeneity of Space & Law of Conservation of Linear Momentum If m1 & m2 be the masses of the particles A & B which are moving with velocities respectively at any time then according to Newton’s second law of motion

Rotational Invariance of Space & Law of Conservation of Angular Momentum Consider two particles A & B interacting with each other. Let be the position vectors of the particles in coordinate system S. Distance between two particles will be given by The P.E. of interaction between the particles depends upon the scalar distance between two particles i.e.,

Suppose the system is rotated through a small angle Z’ Z Suppose the system is rotated through a small angle then the position vectors will get changed to but according to isotropy of space, the distance between two particles will remain same. A S S’ B Y’ O Y X X’ Fig. 1

Continued… Therefore P.E. before rotation is equal to P.E. after rotation Applying Taylor’s theorem Substituting the value to eq. (1), we get

Continued… We know Hence equation (3) becomes We know, when coordinate system is rotated through a vector angle then, where dr is an arc which subtends a vector angle at the origin. Z O Y X

Implicit & Explicit Time Dependence Suppose we are looking at the movement of a classical particle. The relevant variables here are position x(t) and momentum p(t). For example, angular momentum L⃗=x⃗×p⃗ . Since x and p depend on the time, L also depends on time, but in this case it does so only because x and p depend on time. We have basically a function L=L(x,p) which then becomes L(x(t),p(t)). This is because in the definition of L , the time does not play a role. Therefore, we say that this quantity has only an implicit time dependence. In particular, ∂L/∂t=0 . If, however, derived quantity f is defined such that the time occurs explicitly in the definition, for example a=∂v/∂t then a=a(v,t) there is direct dependence (explicit time dependence) on time & ∂a/∂t will not be equal to zero.

Homogeneity of flow of Time & Law of Conservation of Energy We know, force is related to P.E. by We know gravitational force between two masses m1 & m2 is given by It is clear from the expression that force does not depend on time explicitly (directly). Therefore P.E. will also not depend on time explicitly.

Continued… Also K.E. will also not depend on time explicitly but depends on time implicitly Total energy of system E=T+U where E is function of r & t. i.e. E=E(r,t)

Continued…

Frames of Reference

Inertial & Non-Inertial Frames Inertial Reference Frame: Any frame in which Newton’s Laws are valid! Any reference frame moving with uniform motion (non- accelerated) with respect to an “absolute” frame “fixed” with respect to the stars. Non-Inertial Reference Frame: Any frame in which Newton’s Laws are not valid! Any reference frame moving with non-uniform motion (accelerated) with respect to an “absolute” frame “fixed” with respect to the stars.

The most common example of a non-inertial frame = Earth’s surface! We usually assume Earth’s surface is inertial, when it is not! A coord system fixed on the Earth is accelerating (Earth’s rotation + orbital motion) & is thus non-inertial! For many problems, this is not important. For some, we cannot ignore it!

RELATIVITY Problems with Classical Physics Classical mechanics are valid at low speeds But are invalid at speeds close to the speed of light 26

Relativity a special case of the general theory of relativity for measurements in reference frames moving at constant velocity. predicts how measurements in one inertial frame appear in another inertial frame. How they move wrt to each other. 27

Reference Frames The problems described will be done using reference frames which are just a set of space time coordinates describing a measurement. eg. z y x t 28

Galilean-Newtonian Relativity According to the principle of Newtonian Relativity, the laws of mechanics are the same in all inertial frames of reference. i.e. someone in a lab and observed by someone running. 29

Galilean-Newtonian Relativity Galilean Transformations 30

Galilean Transformations We will consider effect of uniform motion on different quantities & laws of physics. We will establish a relationship between the space & time coordinates in two inertial frames of reference. The basic relations were obtained by Galileo & are known as Galilean Transformation Equations. allow us to determine how an event in one inertial frame will look in another inertial frame. assume that time is absolute. 31

Galilean Transformations When second frame is moving relative to first along positive direction of X-axis. In S an event is described by (x,y,z,t). How does it look in S′? z x t z′ x′ t′ vt v O′ O S S’ P(x,y,z.t) (x’,y’z’.t’) Y Y’ 32

Galilean Transformations We will assume that time of occurrence is same in both the frames t = t′ From the diagram, And there is no relative motion in Y & Z-directions z x t z′ x′ t′ vt v O′ O Y Y’ 33

So the Galilean transformations are Inverse Galilean transformations are

Galilean Transformations for Velocity Velocities can also be transformed. Using the previous equations, addition law for velocities 35

Galilean Transformations for Acceleration 36

Galilean Transformations for Force The mass is a constant quantity in Newtonian Mechanics and does not depend on state of rest or motion of the object. According to Newton’s second law of motion w.r.t frame of reference S & S’ But acceleration is invariant under Galilean Transformations

Galilean Transformations Transforming Lengths 38

Galilean Transformations How do lengths transforms transform under a Galilean transform? Note: to measure a length two points must be marked simultaneously. 39

Galilean Transformations v S′ S XA XB Consider the truck moving to the right with a velocity v. Two observers, one in S and the other S′ measure the length of the truck. S is an observer at rest wrt to the earth S′ is an observer on the truck 40

Galilean Transformations v S′ S XA XB The observer in S′ sees the measurements at different points to S In the S frame, an observer measures the length = XB-XA In the S′ frame, an observer measures the length = X′B-X′A 41

Galilean Transformations v S′ S XA XB We know x’=x-vt each point is transformed as follows: Using the Galilean transformation 42

Galilean Transformations U S′ S XA XB Therefore we find that Hence for a Galilean transform, lengths are invariant for inertial reference frames. 43

Important consequence of a Galilean Transformations All the laws of mechanics are invariant under a Galilean transform. 44

Galilean Transformations When second frame is moving relative to first along a straight line along any direction In S an event is described by (x,y,z,t). How does it look in S′? Z’ P(x,y,z.t) (x’,y’z’.t’) S’ Z S Y’ O’ O Y X’ X

S & S’ are two frames of references. S’ is moving w.r.t. S with velocity ‘v’ The coordinates of point P are (x,y,z,t) & (x’,y’,z’,t’) relative to S & S’ respectively. Time is measured from the instant at which the two origins O & O’ coincide Then after time t, O’ is separated from origin O by displacement Then

Applying the triangle law of vectors in triangle OO’P, we get Now Therefore eq. 1 becomes

Invariance of Law of Conservation of Linear momentum Under Galilean Transformations, velocity is not invariant, therefore linear momentum is also not invariant. We have to check that whether law of conservation of linear momentum holds or not According to whether law of conservation of linear momentum “in the absence of external force, the total momentum of the system remains unchanged or conserved”

Consider two particles

Invariance of Law of Conservation of Kinetic Energy In S

Problems with Newtonian- Galilean Transformation Are all the laws of Physics invariant in all inertial reference frames? Problems with Newtonian- Galilean Transformation 51

Problems with Newtonian- Galilean Transformation Are all the laws of Physics invariant in all inertial reference frames? For example, are the laws of electricity and magnetism the same? Problems with Newtonian- Galilean Transformation 52

Problems with Newtonian- Galilean Transformation Are all the laws of Physics invariant in all inertial reference frames? For example, are the laws of electricity and magnetism the same? For this to be true Maxwell's equations must be invariant. Problems with Newtonian- Galilean Transformation 53

Problems with Newtonian- Galilean Transformation From electromagnetism we know that, Since and are constants then the speed of light is constant . 54

Problems with Newtonian- Galilean Transformation From electromagnetism we know that, Since and are constants then the speed of light is constant . However from the addition law for velocities 55

Problems with Newtonian- Galilean Transformation Therefore we have a contradiction! Either the additive law for velocities and hence absolute time is wrong Or the laws of electricity and magnetism are not invariant in all frames. 56

Fictitious Forces Some problems (like rigid body rotation!): Using an inertial frame is difficult or complex. Sometimes its easier to use a non-inertial frame! “Fictitious Forces”: If we are careful, we can the treat dynamics of particles in non-inertial frames. Start in inertial frame, use Newton’s Laws, & make the coordinate transformation to a non-inertial frame. Suppose, in doing this, we insist that our equations look like Newton’s Laws (look like they are in an inertial frame).  Coordinate transformation introduces terms on the “ma” side of F = mainertial. If we want eqtns in the non-inertial frame to look like Newton’s Laws, these terms are moved to the “F” side & we get: “F” = manoninertial. where “F” = F + terms from coord transformation  “Fictitious Forces” !