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Space-Time Symmetry

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Published in: Physics
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Space-Time symmetry 

Akhilesh K / Lucknow

4 years of teaching experience

Qualification: M.Sc (NIT Rourkela - 2019)

Teaches: All Subjects, English, Mathematics, Science, Chemistry, Physics, Algebra, IIT JEE Mains, AIPMT, NEET

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  1. WWÅSö11Å, ew!l-eoeds
  2. 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 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.
  3. 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 F = xi +yj+zk in right handed coordinate system then in left handed coordinate system -xi yj-zk However laws of physics (e.g. F=ma) remains invariant.
  4. 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.
  5. 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.
  6. Taylor's Theorem If & are two variables, having infinitesimal change dil & dF2 respectively, then according to Taylor's theoremin vector form where VI f &V2f
  7. 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. B c
  8. Figure 11.1 Primed coordinate system S' is in translational motion with respect to a fixed stationary unprimed coordinate system S.
  9. 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
  10. N/A
  11. Homogeneity of Space & Newton's Third law of motion According to Principle of Homogeneity ...U) From Taylor' s theoremin vector form + dF,F2 + = ) + + dF.V where f & V2f — —i + dF.(VlU + since O ...(2) but for conservative forces & É21 = —V2U . eq. (2) becomes which is Newton' s Third Law.
  12. Homogeneity of Space & Law of Conservation of Linear Momentum If ml & be the masses of the particles A& B which are moving with velocities & 132 respectively at any time then according to Newton's second law of motion & m2ä2 adding (4) & (5) dil dF2 dt dt dil dt dF2 dt according to Newton's third law ( eq. (3) ) 12 21 dil dF2 dt dt + nt2F2) = constant d (mlF1) dt d (m2F2) dt dt which is law of conservation of linear momentum.
  13. 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.,
  14. z' s' z s Fig. 1 Suppose the system is rotated through a small angle dQ then the position vectors will get changed to Fl+dFl &F2+dF2 but according to isotropy of space, the distance between two particles will remain same.
  15. Continued... Therefore P.E. before rotation is equal to P.E.U(FI +dFl,F2 +dF2) after rotation Applying Taylor's theorem U(FI + + = ) + + ' Substituting the value to eq. (1), we get -k dF2.V2U = O
  16. Continued... We know F = -VU = & F 21 ' Hence equation (3) becomes -dFl.F12 -dF2.F21 = O -V2U z We know, when coordinate system is rotated through a vector angle dÖ then, x where dr is an arc which subtends a vector angle dö at the origin. F2Xdö ...(7)
  17. from Newton' s Second law dil dP2 ...(8) 12 dt dt Substituting eq. 6,7 & 8 in eq. ( .dP2 = O dt dt d(F1 x DI) dil dil Now dt dt dt dil but = 0 dt dt Similarly —rp< dt ...UI) dt Interchanging Dot & Cross product dt dil dQ.F1 x hence eq.(9) becomes d(F2xP2) dt dt sincedQ 0 dt dt dpi x = O dt dt dt dP1 x dP2 = 0 dt dt but angular momentumis given by + 0 + L) = constant dt
  18. 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 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, öL/öt=O . If, however, derived quantity f is defined such that the time occurs explicitly in the definition, for example a=öv/öt then a=a(v,t) there is direct dependence (explicit time dependence) on time & öa/öt will not be equal to zero.
  19. Homogeneity of flow of Time & Law of Conservation of Energy , We know, force is related to P.E. by ...U) , We know gravitational force between two masses ml & is given by Gnqm2 , 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. (2)
  20. Continued... Also K.E. will also not depend on time explicitly but depends on time implicitly K.E. T=-mv 2 (3) Dt Total energy of system E=T+U where E is function of r & t. i.e. E=E(r,t)
  21. Continued... Dt Dr Dv but Dr -F — mv Dr 2 Dr Dt ....(5) 2 Dr — mv —dr + Dr using eq. (2) & (3), Dr Dt Dt dt m 2 at — ma (6)
  22. substituting eq. (5) & (6) in eq. (4) dr = (ma - F) dt dt but F — ma dt — constant

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