Minkowski force

According to our discussion of the 4-velocity in the last section the natural covariant generalization of Eq. (3.123) is given by 

126) holds for any inertial frame of reference, and thus it is also true in IS where v = 0, which immediately yields 

Insertion of this result in the equation of motion (3.124) yields the Minkowski 

The Minkowski force / in any other inertial frame IS, which moves with velocity —u relative to IS, may therefore be obtained via a Lorentz transformation. 

For the simple case of a Lorentz boost in x-direction with v = ve the relation between the Newtonian force F = (fi, F , fs) and the Minkowski force is by virtue of Eq. (3.66) given by 

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For the general case of an arbitrary Lorentz boost A( ) we note that the inverse transformation A(—i ) is easily obtained from Eq. (3.81) by switching the signs of the velocity components V (cf. Eq. (3.82)). The Minkowski force / in IS is therefore given by... 

As the most important example of a Minkowski force in special relativity we will now explicitly consider the Lorentz force on a charged particle, which has already been encountered in section 2.4. In IS, where the particle is at rest, it experiences a force... 

Note that we have artificially extracted the electric field in the last equality of Eq. (3.134), which represents an exact relation. Inserting this expression for the electric field into the relation for the Minkowski force as given by Eq. (3.131) directly yields the four-dimensional Lorentz force / in IS,... 

These three equations have to equal the spatial components of the elementary equation of motion as given by Eq. (3.124), which for a free particle with vanishing Minkowski force f = 0 read... 

The rest mass remains constant if the 4-force and 4-velocity are orthogonal in the Minkowski sense that X u, = 0. 

Theories like those of Lemaitre or Friedmann, which predict an expanding universe, are all based on forcing an affine metric, such as the Robertson-Walker metric, on the projective geometry of space-time. This has the effect of splitting local Minkowski space into separate space and time coordinates, without the natural complex relationship that ties space and time together. 

General relativity is the theory that gave physical content to Riemaim s formulation of curved mathematical space and identifies the four-dimensional metric tensor with the gravitational field. The four dimensions of general relativity are the same as in the Minkowski space of special relativity. The velocity of light remains a constant in free space and the inability to specify simultaneous events remains in force. 

The Vanishing of Apparent Forces The Galilean Transformation The Michelson-Morley Experiment The Galilean Transformation Crashes The Lorentz Transformation New Law of Adding Velocities The Minkowski Space-Time Continuum How do we Gel E =... 

The radiation field exerts direct forces and torques on condensed matter via the Minkowski stress [64]. Radiation pressure 11, is on the order of If, DE k, I/c k, P/Ac where D is the electric displacement, I is the irradiance, c is the speed of light, P is the incident power, and A is the area. If the direct optical force is to cause a deformation, the radiation pressure would have to overcome the aligning effect of... 

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