Classical Mechanics and Special Relativity with GA

1. Classical Mechanics and Special Relativity with GA Suprit Singh

2. An Outline of Geometric Algebra A little Un-Learning Geometric ProductClifs Vector Algebra Vector Equations • GA is Clifford Algebra with GEOMETRIC and PHYSICAL interpretation of its mathematical elements. • We need to unlearn a few ambiguous things ironically prevalent : the product of vectors… • The magnitude of a vector is contained in the scalar product… • The direction being specified by Cross product….which is one roadblock we wish to clear… so stamp them out..introduce the ‘OUTER’ product

3. An Outline of Geometric Algebra A little Un-Learning Geometric Product Clifs Vector Algebra Vector Equations Recipe for the Vector Salad Define The addition of two is then Perfectly legitimate powerful axiom…. any resemblance???? Yes...its like a complex number, isn’t it? We’ll use this from now on, instead of the Inner and Outer products…

4. An Outline of Geometric Algebra A little Un-Learning Geometric Product Clifs Vector Algebra Vector Equations The elements of GA are generated through exterior products..scalars, vectors, bivectors, trivectors and so on…multi-vectors…multi-multi-vectors…and we can sum any of them…(remember its not the ordinary addition) such a general element we call a ‘CLIF’ and they form a linear space. With all this..there’s a wonderful surprise package..you can ‘divide’ here… Here’s a example ,

5. An Outline of Geometric Algebra A little Un-Learning Geometric ProductClifs Vector Algebra Vector Equations • Okay, here are some of the ‘Old’ things in ‘New’ and ‘Better’ ways… • The area of a parallelogram, with the orientation • Vector Identities

6. An Outline of Geometric Algebra A little Un-Learning Geometric ProductClifs Vector Algebra Vector Equations • And some of the New Division Flavor

7. The Vector and Spinor PlaneG(2) Plane- SpinorsRotation in a plane Schwarzschild Black Hole • The Coordinates Metric Singularities Falling In Formation Penrose Diagram • The N-dimensional Euclidean Space is the solution of • where the dimensionality of the Space is hidden in the element I, called the pseudoscalar. • Choose • We get the solution • which corresponds to the Euclidean Plane spanned by

8. The Vector and Spinor PlaneG(2)Plane- SpinorsRotation in a plane • Interpreting the pseudoscalar: • Directed Unit Area in the plane • Generator of Rotations • Plus we have our Algebra splitting two • The even sub-algebra forms the well known Complex Plane

9. The Vector and Spinor PlaneG(2) Plane- SpinorsRotation in a plane • The re exists one-to-one mapping between Complex Numbers and Vectors through the choice of scalar axis • We call them 2-spinors as their ‘Operational ‘ Job is Rotating vectors, see… • This also then implies that Angle better be interpreted as area

10. The Three Space An Extended Choice Cross ProductQuaternions Reflections Rotations • Extending our previous choice of I, • we have 8-d graded linear space • We got 3 bivectors corresponding to 3 planes…so alls same as 2-plane…with a few extras

11. The Three Space An Extended ChoiceCross Product Quaternions Reflections Rotations • OMG, Where’s the Cross Product ???? • Here it is…the dual of the plane formed by two vectors… • And here’s where the Quaternions materialize… • They are bivectors and not the vectors if the scalar part is set to zero…this solves their Reflection problem….Trumpets Please...

12. The Three Space An Extended Choice Cross ProductQuaternions Reflections Rotations • Here’s the First Power Display of GA…The reflection of a vector written compactly as a simple expression • And for any general multivector

13. The Three Space An Extended Choice Cross ProductQuaternions Reflections Rotations • The Second Power : Rotations expressed in a double sided generic form… • Start off with a vector and subject it to two successive reflections : • Define Rotor, R and then inspecting components of vector, a in and out of plane A..

14. The Three Space An Extended Choice Cross ProductQuaternions Reflections Rotations We get to an Important conclusion :

15. The Three Space An Extended Choice Cross ProductQuaternions Reflections Rotations • Euler Angles : We require • Hence adopt the procedure :

16. The Three Space An Extended Choice Cross ProductQuaternions Reflections Rotations A sleek and simple representation

17. Spacetime Algebra (STA)Vectors Higher Elements Spacetime Split Dynamical Relative Vectors Lorentz Rotation • The postulates of Special Relativity require a non-Euclidean metric signature… • That is, we need modify our condition in Euclidean spaces.. • The mixed metric gives rise to reciprocal spaces… • Then any spacetime point is given by :

18. Spacetime Algebra (STA)Vectors Higher ElementsSpacetime Split Dynamical Relative Vectors Lorentz Rotation • Bivectors : • The timelike bivectors square to +1…we can see hyperbolic geometry coming up.. • Pseudoscalar : Gives a useful map between the 2 types of Bivectors…

19. Spacetime Algebra (STA)Vectors Higher Elements Spacetime Split Dynamical Relative Vectors Lorentz Rotation • The invariant interval for a timelike path for λ = τ implies… • In the rest frame, the proper time is a preferred parameter of path • such that velocity is timelike..and can be identified with • Choose the frame of rest vectors normal to v…den a general event can be decomposed as • where as you see..the x is a relative vector/ spacetime bivector for a event… • The invariants remain same for all observers…

20. Spacetime Algebra (STA)Vectors Higher Elements Spacetime Split Dynamical Relative Vectors Lorentz Rotation • Some Relative Vectors… • Relative Velocity : • Momentum and Energy : • Proper Acceleration Bivector:

21. Spacetime Algebra (STA)Vectors Higher Elements Spacetime Split Dynamical Relative Vectors Lorentz Rotation • Generalize 3D Euclidean rotation to Minkowski…requiring… • defining the Lorentz transformation from one frame to another…through 6 generators.. • which for example for boost in z-direction gives…

22. Spacetime Algebra (STA)Vectors Higher Elements Spacetime Split Dynamical Relative Vectors Lorentz Rotation • Some applications… • Relativistic Velocity Addition • Doppler Effect

23. Classical Mechanics Constant Force Angular Momentum Central Force Non-inertial frames Magnetic Field • Motion under a constant force :

24. Classical Mechanics Constant ForceAngular Momentum Central Force Non-inertial frames Magnetic Field • Angular momentum Bivector : encodes are swept by a radius vector around some origin… • Hence the definition requires.. • The toque is also then • In terms of the geometric product, • In situations where there is spherical symmetry, L is conserved, with magnitude

25. Classical Mechanics Constant Force Angular MomentumCentral Force Non-inertial frames Magnetic Field • V=V(r), then Total E is conserved… • Consider • Spinor Way:

26. Classical Mechanics Constant Force Angular MomentumCentral Force Non-inertial framesMagnetic Field

27. Classical Mechanics Constant Force Angular MomentumCentral Force Non-inertial frames Magnetic Field Motion in a constant Magnetic Field :

28. A Look at ElectromagnetismThe Lorentz Force Covariant Maxwell’s Equation • The quantity on left is relative vector, • , we multiply both sides by ‘gamma’ to get derivative wrt proper time… • Similarly, for B, • Combining , • where : • Define:

29. A Look at ElectromagnetismThe Lorentz ForceCovariant Maxwell’s Equation • The Lorentz Force law can now be written as, • The power equation is now, • Adding latter to the first after a suitable multiplication, we have the covariant law… • The Covariant Maxwell Equation :

30. A Look at ElectromagnetismThe Lorentz Force Covariant Maxwell’s Equation • First, we combine the two equations for E and B as • Introducing F : • Writing • We have the Maxwell Equation : • And as a consequence,

31. Thank you