- 2. Introduction Basic Postulates of Special Theory of Relativity. Lorentz Transformation Velocity Addition and Thomas Precession
- 3. At the end of nineteenth century, there are two mismatched descriptive phenomena Newtonian Mechanics Maxwell’s Electromagnetic Theory Newtonian Mechanics assumed that Maxwell’s wave equation gave Any frame moving at constant velocity with respect to an inertial frame is also an inertial frame.
- 4. Consider two frames denoted by 𝑆 𝑎𝑛𝑑 𝑆′ with 𝑡, 𝑥, 𝑦, 𝑧 𝑎𝑛𝑑 (𝑡′ , 𝑥′ , 𝑦′ , 𝑧′ ) the coordinates in 𝑆 𝑎𝑛𝑑 𝑆′ , respectively. The coordinate axes are aligned and, 𝑥 𝑎𝑙𝑜𝑛𝑔 𝑥′, and so on. Let 𝑆′ be moving relative 𝑆 in the +x-direction at a speed 𝑣, 𝑎𝑠 𝑠ℎ𝑜𝑤𝑛 𝑖𝑛 𝑓𝑖𝑔.
- 5. The spacetime coordinates of 𝑆 and 𝑆′ are related by simple expressions
- 6. Transformation of this type is called Galilean transformations. Under this assumption, it follows that Newton’s Second Law Relating to the applied force (𝑭) and the momentum (𝒑) remains invariant, and The time in both 𝑆 and 𝑆′ frames is assumed to be 𝒕 = 𝒕′ .
- 7. The Newtonian world view is that “The universe consists of three spatial directions and one time direction.” All the observers agree on the time direction up to a possible choice of units. Under these assumptions, there are no universal velocities. If u and u’ are the velocities of a particle as measured in two frames moving with relative velocity v, then
- 8. On the other hand, Maxwell’s Electromagnetic equations, there is a universal velocity 𝒄 , which is interpreted as speed of light. Maxwell universal Velocity is inconsistent with the Newtonian’s view having no universal velocity. Then either Newtonian Mechanics or Maxwellian mechanics would have to be modified. Then Albert Einstein came with Special theory of relativity decided that Maxwell is right.
- 9. Einstein’s Special Theory of Relativity Postulates are: 1. The laws of physics are the same to all inertial observers. 2. The speed of light is the same to all inertial observers. This means that time is now not an invariant quantity, it is now covariant. So in order to relate time with the space coordinates there must be some conversion factor which is 𝑐𝑑𝑡 . In SI Systems of units, 𝑐𝑑𝑡 has dimensions of meters. Hence, space and time in special theory of relativity is a single entity and we called is spacetime. This spacetime is the geometric framework within which we perform physics.
- 10. The square of distance in that space time, △ 𝑠2 , between two points A and B is given by If the separation is infinitesimal, the △ is replaced by the differential symbol 𝑑. Such a point in spacetime is known as event. The term event is used because such a point has a definite location and definite time in any frame. ……….. (1)
- 11. The above equation can be written as Objects that travels on timelike path are called tardyons. Hypothetical bodies that travels on spacelike path are called tachyons, and the objects moving with speed of light are called lightlike. ...........(1)a
- 12. In the limits of small displacements in Cartesian coordinate system, The four-dimensional space with an interval defined is often called Minkowski space to distinguish it from a four dimensional Euclidean space for which there would be no minus sign. ………..(2)
- 13. Let 𝑆 𝑎𝑛𝑑 𝑆′ are two different inertial frames, then Now, the time coordinate can no longer stand independent of transformation. Now the time measured in a Laboratory frame is different from that measured by an observer at rest with respect to the body under study. z y x x' y' z' vS S’ ………(3)
- 14. Both times are not same so we distinguish them by calling “the time interval measured by a clock at rest with respect to a body is the proper time, while the other inertial observer uses a time that is called Laboratory time. Consider the relation between the proper time 𝜏, measured by observer at rest with coordinates (𝑡, 𝑥, 𝑦, 𝑧) and the Laboratory time 𝑡, which is moving with velocity 𝒗 having coordinates (𝜏, 𝑥′, 𝑦′, 𝑧′). By Eq. (2) and (3) This shows that 𝑑𝑡 > 𝑑𝜏. This effect is called “Time Dilation”. Moving clock appears to run slower.
- 15. The invariance of the interval expressed in Eq. (3) naturally divides space-time into three regions relative to any event A at time tA (A is located at 𝑥 = 𝑦 = 𝑡 = 0 in Fig.).