The length of a quaternion in curved spacetime: a close relative of the affine parameter of general relativity

doug  <sweetser@world.com>

The affine parameter of general relativity
Length in flat spacetime
Length in curved spacetime
Implications

The affine parameter of general relativity

The affine parameter is defined in Misner, Thorne and Wheeler as a multiple of the proper time plus a displacement.

λ  =  a τ  +  b;

λ  =  a τ  +  b

The affine parameter is used to determine length in curved spacetime.  In this notebook, the length of a quaternion in curved spacetime will be analyzed.  Under certain approximations, this length will depend on the square of the affine parameter, but the two measures are slightly different.

Length in flat spacetime

Calculating the square of the interval between two events in flat spacetime was straightforward: take the difference between two quaternions and square it.

\(q\_1\  = \ q[t\_1, x\_1, y\_1, z\_1]; 

    q\_2\  = \ q[t\_2, x\_2, y\_2, z\_2]; 

    \((L\_flat = 
          Simplify[
	
            \((\((q\_2 - \ q\_1)\).\((q\_2 - \ q\_1)\))\) /. \ toDelta])\).{
        1, 0, 0, 0}\){(((Δ  t))^2)-(((Δ  x))^2)-(((Δ  y))^2)-(((Δ  z))^2),
	2 (Δ  t) (Δ  x),2 (Δ  t) (Δ  y),2 (Δ  t) (Δ  z)}

(Subscript[L, flat]) = 
      (((q  -  (q^′)))^2) =  
        ((dt^2)  -  d  ((Overscript[X, ⇀])^2),  2 dt d  (Overscript[X, ⇀]))

The first term is the square of the interval.  Spacetime is flat in the sense that the first term is exactly like the Minkowski metric in spacetime. There are quaternions which preserve the interval, and those quaternions were used to solve problems in special relativity.

Although not important in this context, it is significant that the value of the vector portion depends upon the observer.  This gives a way to distinguish between various frequencies of light for example.

Length in curved spacetime

Consider if the origin  is located at two different locations in spacetime.  Characterize each origin as a quaternion, calling the o and o'.  In flat spacetime, the two origins would be identical.  Calculate the interval as done above, but account for the change in the origin.

\(\((L\_curved = 
          Simplify[
	
            \((\((\((\((q\_2 + \ origin\_2)\) - \ \((q\_1 + \ origin\_1)\))\).
                      
			\((
                        \((q\_2 + \ origin\_2)\) - \ \((q\_1 + \ origin\_1)\))
                        \))\)
			 /. \ toDelta)\) /. \ toDelta])\).{1, 0, 
        0, 0}\){(((Δ  ot+Δ  t))^2)-(((Δ  ox+Δ  x))^2)-(((Δ  oy+Δ  y))^2)-(((Δ  oz+Δ  z))^2),
	2 (Δ  ot+Δ  t) (Δ  ox+Δ  x),
	2 (Δ  ot+Δ  t) (Δ  oy+Δ  y),
	2 (Δ  ot+Δ  t) (Δ  oz+Δ  z)}

\(L\_curved = 
      \(\((\((q + o)\)\  - \ \((q\^′ + o\^′)\))\)\^2
 = 
        \((d \((t + t\_o)\)\^2\  - \ 
            d \((X\&⇀ + X\&⇀\_o)\)\^2, \ 
          2\ d \((t + t\_o)\)\ d 
            \((X\&⇀ + X\&⇀\_o)\))\)\)\)

Examine the first term more closely by expanding it.

Expand[(Subscript[L, curved])[[1, 1]]](((Δ  ot))^2)-(((Δ  ox))^2)-(((Δ  oy))^2)-(((Δ  oz))^2)+
2 (Δ  ot) (Δ  t)-2 (Δ  ox) (Δ  x)-2 (Δ  oy) (Δ  y)-2 (Δ  oz) (Δ  z)+
(((Δ  t))^2)-(((Δ  x))^2)-(((Δ  y))^2)-(((Δ  z))^2)

((dt^2) - d  ((Overscript[X, ⇀])^2))  +  
      (((Subscript[dt, o])^2) - d  ((Subscript[(Overscript[X, ⇀]), o])^2)) +  2 dt (Subscript[dt, o])  - 
      2 d  (Overscript[X, ⇀])  d  (Subscript[(Overscript[X, ⇀]), o])

The length in curved spacetime is the square of the interval (invariant under a boost) between the two origins, plus the square of the interval between the two events, plus a cross term, which will not be invariant under a boost.  The length is symmetric under exchange of the event with the origin translation.

Lcurved looks similar to the square of the affine parameter:

Expand[(λ^2)](b^2) + 2 a b τ + (a^2) (τ^2)

(λ^2) = (b^2) + 2 a b τ + (a^2) (τ^2)

In this case, b^2 is the origin interval squared and a = 1.  There is a difference in the cross terms.  However, in the small curvature limit, delta to >> delta Xo, so tau ~ delta to.  Under this approximation, the square of the affine parameter and Lcurved are the same.

For a strong gravitational field, Lcurved will be different than the square of the affine parameter.  The difference will be solely in the nature of the cross term.  In general relativity, b and tau are invariant under a boost.  For Lcurved, the cross term should be covariant.  Whether this has any effects that can be measured needs to be explored.

There exist quaternions which preserve Lcurved because quaternions are a field (I haven't found them yet because the math is getting tough at this point!)  It is my hope that those quaternions will help solve problems in general relativity, as was the case in special relativity.

Implications

A connection to the curved geometry of general relativity was sketched.  It should be possible to solve problems with this "curved" measure.  As always, all the objects employed were quaternions.  Therefore any of the previously outline techniques should be applicable.  In particular, it will be fun in the future to think about things like

\(\((L\_curved = 
          Simplify[
	
            \((\((Transpose[
						
                        \((\((q\_2 + \ origin\_2)\) - \ 
                            \((q\_1 + \ origin\_1)\))\)].
			
                      \((\((q\_2 + \ origin\_2)\) - \ 
                          \((q\_1 + \ origin\_1)\))\))\)
			 /. \ toDelta)
                \) /. \ toDelta])\).{1, 0, 0, 0}\){(((Δ  ot+Δ  t))^2)+(((Δ  ox+Δ  x))^2)+(((Δ  oy+Δ  y))^2)+(((Δ  oz+Δ  z))^2),0,0,0}

\(\(\((\((q + o)\)\  - \ \((q\^′ + o\^′)\))\)\^*\) 
        \((\((q + o)\)\  - \ \((q\^′ + o\^′)\))\)
 = 
      \(\((d \((t + t\_o)\)\^2\  + \ 
            d \((X\&⇀ + X\&⇀\_o)\)\^2, \ 
          2\ d \((t + t\_o)\)\ d 
            \((X\&⇀ + X\&⇀\_o)\))\)
 = 
        \((\((dt\^2 - d  X\&⇀\^2)\)\  + \ 
            \((dt\_o\^2 - d  X\&⇀\_o\^2)\) + \ 2\ dt\ dt\_o\  + 
            2\ d  X\&⇀\ \ d  X\&⇀\_o, \ ...)\)\)\)

which could open the door to a quantum approach to curvature.