Classical Mechanics and Special Relativity with GA Suprit Singh.

Classical Mechanics and Special Relativity with GA

Suprit Singh

• 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

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…



• 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 ,



• Okay, here are some of the ‘Old’ things in ‘New’ and ‘Better’ ways…

– The area of a parallelogram, with the orientation

– Vector Identities



• And some of the New Division Flavor



The Coordinates Metric Singularities Falling In Formation Penrose Diagram

Schwarzschild Black Hole

• 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

The Vector and Spinor Plane

G(2) Plane- Spinors Rotation 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



• 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



The Three Space

An Extended Choice Cross Product Quaternions 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

• 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...

The Three Space


• Here’s the First Power Display of GA…The reflection of a vector written compactly as a simple expression

• And for any general multivector

The Three Space


• 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..

The Three Space


The Three Space


We get to an Important conclusion :

• Euler Angles : We require

• Hence adopt the procedure :

The Three Space


The Three Space


A sleek and simple representation

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 :



• 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…

• 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…



• Some Relative Vectors…

Relative Velocity :

Momentum and Energy :

• Proper Acceleration Bivector:



• 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…



• Some applications…

• Relativistic Velocity Addition

• Doppler Effect



• Motion under a constant force :

Classical Mechanics

Constant Force Angular 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

Classical Mechanics


• V=V(r), then Total E is conserved…

• Consider

Classical Mechanics


Spinor Way:

Classical Mechanics


Classical Mechanics


Motion in a constant Magnetic Field :

• 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:

A Look at Electromagnetism

The Lorentz Force Covariant 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 :



• First, we combine the two equations for E and B as

• Introducing F :

• Writing

• We have the Maxwell Equation :

And as a consequence,

Thank you

Classical Mechanics and Special Relativity with GA Suprit Singh.

Documents

Transcript of Classical Mechanics and Special Relativity with GA Suprit Singh.