The Geometry Of Space-time With Broken Lorentz Symmetry

Local Lorentz invariant and CPT invariant are the bases of modern physics, and modern physics experiments have given a serious restriction on it. But nowadays, there are more and more indicators shows that one or both of they might be broken. So, find a new way to contain the symmetry broken and combine it with the all the vested physics facts might be a meaningful work.Since 2006, Cohen and Glashow shown the physics idea about broken Lorentz symmetry in [1], there are lots of new physical construction have been worked out. This article try to combine the broken Lorentz symmetry and Finslerian geometry, find out the connection between them, and pose the physics rule on finslerian geometry.Firstly, we set off from the symmetry broken. We select the four-element sub Lie algebra and three-element sub Lie algebras of Lorentz algebra to be the original local symmetry. And then calculate the semidirect products of them and the translation algebra of (3,1) space-time. After that, deform the semidirect products, and then we get the symmetry of the Lorentz-broken space-time.Secondly, find out the matrix presentation of the deformed symmetry. After chosen an appropriate matrix presentation, we can construct the invariant finslerian measurement function under the group action by Killing formula. This measurement can reflect the Lorentz-broken symmetry of the geometry.After that, we can construct the action of point-like particle by finslerian measurement, and then get the dispersion relation of it.At this part, we will see that the Finslerian space-time can be understood as a flat Riemannian space-time with some kind of special interaction potential. So, we can think that Finslerian physics is some kind of dynamical model of special interaction.At the last, we will find out the action of scalar field, spinor field and vector field, especially the gauge field by the Finslerian measurement and the dispersion relation of point-like particles. And if we require the gauge invariant of action, then we will get the gauge interaction rule of the Finslerian space-time.Also, we will see that the field in Finslerian space-time can behave very different. The action of them are almost all nonlocal, and this kind of nonlocal property can be understood as some kind of "renormalization" of field and the mass of field. In some kind of special Finslerian space-time, the Finslerian geometry will give the gauge field "mass".This article just discusses some problem in the frame of classical physics, not contain the subject of quantization. It can be discussed in some kind of same way.The use of Finslerian geometry in physics is still very limited. Via the tiny attempt in this article, we can see that physics in Finslerian geometry can behave very novelty and interesting, and some of Finslerian geometry might show us some very usefully new physics. With the develop of Finslerian geometry in physics, we would find out more and more exciting new physics behavior.