Gravity, gauge theories and geometric algebra (original) (raw)

Abstract

A new gauge theory of gravity is presented. The theory is constructed in a flat background spacetime and employs gauge fields to ensure that all relations between physical quantities are independent of the positions and orientations of the matter fields. In this manner all properties of the background spacetime are removed from physics, and what remains are a set of

Figures (10)

Table 1: The symmetries of the action integral (3.55). The Coupled Dirac Equation Having arrived at the action (3.55) we now derive the coupled Dirac equation by extremising with respect to w, treating all other fields as external. When applying the Euler-Lagrange equations to the action (3.55) the w and w fields are not treated as independent, as they often are in quantum theory. Instead, we just apply the rules for the multivector derivative discussed in Section 2.2 and Appendix B. The Euler-Lagrange equations can be written in the form

Table 1: The symmetries of the action integral (3.55). The Coupled Dirac Equation Having arrived at the action (3.55) we now derive the coupled Dirac equation by extremising with respect to w, treating all other fields as external. When applying the Euler-Lagrange equations to the action (3.55) the w and w fields are not treated as independent, as they often are in quantum theory. Instead, we just apply the rules for the multivector derivative discussed in Section 2.2 and Appendix B. The Euler-Lagrange equations can be written in the form

a IR I I a a a A summary of our definitions and conventions is contained in Table 2. We have ndeavoured to keep these conventions as simple and natural as possible, but a word s in order on our choices. It will become obvious when we consider the variational rinciple that it is a good idea to use a separate symbol for the spacetime vector lerivative (V), as opposed to writing it as 0,. This maintains a clear distinction etween spacetime derivatives, and operations on linear functions such as ‘contrac- ion’ (0,:) and ‘protraction’ (0,/A). It is also useful to distinguish between spinor nd vector covariant derivatives, which is why we have introduced separate D and ) symbols. We have avoided use of the d symbol, which already has a very specific neaning in the language of differential forms. Finally, it is necessary to distinguish yetween rotation-gauge derivatives (D,) and the full covariant derivative with the -field included (a-D). Using D, and a-D for these achieves this separation in the implest possible manner.

a IR I I a a a A summary of our definitions and conventions is contained in Table 2. We have ndeavoured to keep these conventions as simple and natural as possible, but a word s in order on our choices. It will become obvious when we consider the variational rinciple that it is a good idea to use a separate symbol for the spacetime vector lerivative (V), as opposed to writing it as 0,. This maintains a clear distinction etween spacetime derivatives, and operations on linear functions such as ‘contrac- ion’ (0,:) and ‘protraction’ (0,/A). It is also useful to distinguish between spinor nd vector covariant derivatives, which is why we have introduced separate D and ) symbols. We have avoided use of the d symbol, which already has a very specific neaning in the language of differential forms. Finally, it is necessary to distinguish yetween rotation-gauge derivatives (D,) and the full covariant derivative with the -field included (a-D). Using D, and a-D for these achieves this separation in the implest possible manner.

In Section 3.3 we found that the minimally-coupled Dirac action gave rise to the minimally-coupled Dirac equation only when Dyh(d; det(h)~*) = 0. We now see that this requirement amounts to the condition that the spin tensor has zero contraction. But, if we assume that the Q(a) field only couples to a Dirac fermion field, then the coupled Dirac action (3.55) gives

In Section 3.3 we found that the minimally-coupled Dirac action gave rise to the minimally-coupled Dirac equation only when Dyh(d; det(h)~*) = 0. We now see that this requirement amounts to the condition that the spin tensor has zero contraction. But, if we assume that the Q(a) field only couples to a Dirac fermion field, then the coupled Dirac action (3.55) gives

Table 3: Covariant derivatives of the polar-frame unit timelike bivectors.

Table 3: Covariant derivatives of the polar-frame unit timelike bivectors.

Table 4: Equations governing a radially-symmetric perfect fluid. An equation of state and initial data p(r,to) and g2(r,to) determine the future evolution of the system.

Table 4: Equations governing a radially-symmetric perfect fluid. An equation of state and initial data p(r,to) and g2(r,to) determine the future evolution of the system.

Figure 1: Matter and photon trajectories for radial motion in the in the Newtonian gauge. The solid lines are photon trajectories, and the horizon lies at r=2. The broken lines represent possible trajectories for infalling matter. Trajectory I is for a particle released from rest at r = 4. Trajectory II is for a particle released from rest at r=oo.

Figure 1: Matter and photon trajectories for radial motion in the in the Newtonian gauge. The solid lines are photon trajectories, and the horizon lies at r=2. The broken lines represent possible trajectories for infalling matter. Trajectory I is for a particle released from rest at r = 4. Trajectory II is for a particle released from rest at r=oo.

Table 5: Equations governing a radially-symmetric perfect fluid — case with a non- zero cosmological constant A. The shaded equations differ from those of Table 4.

Table 5: Equations governing a radially-symmetric perfect fluid — case with a non- zero cosmological constant A. The shaded equations differ from those of Table 4.

Table 6: Equations governing a homogeneous perfect fluid.

Table 6: Equations governing a homogeneous perfect fluid.

Figure 5: Streamlines of the D field. The horizon is at r = 2 and the charge is placed on the z-axis. The charge is at z = 3 and z = 2.01 for the top and bottom diagrams respectively. The streamlines are seeded so as to reflect the magnitude of D. The streamlines are attracted towards the origin but never actually meet it. Note the appearance of a ‘cardiod of avoidance’ as the charge gets very close to the horizon. The equation for this cardiod is r = M(1+ cos), which is found by setting D=0 whena=2M.

Figure 5: Streamlines of the D field. The horizon is at r = 2 and the charge is placed on the z-axis. The charge is at z = 3 and z = 2.01 for the top and bottom diagrams respectively. The streamlines are seeded so as to reflect the magnitude of D. The streamlines are attracted towards the origin but never actually meet it. Note the appearance of a ‘cardiod of avoidance’ as the charge gets very close to the horizon. The equation for this cardiod is r = M(1+ cos), which is found by setting D=0 whena=2M.

The metric is then given by the 4 x 4 matrix Here S),,” is the torsion tensor, equal to the antisymmetric part of the connection:

The metric is then given by the 4 x 4 matrix Here S),,” is the torsion tensor, equal to the antisymmetric part of the connection:

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