- #1

Logic314

- 10

- 8

I have previously studied special relativity, but only at an introductory level. So I decided to explore the subject more in detail later by thinking and working things out on my own, in addition to doing research online. In particular, I seem to have noticed some intriguing patterns between quantities in special relativity, and based on these, have some conjectures to the mathematical relationships governing these quantities:

I know that a general infinitesimal Lorentz transformation has two components, a rotation component which rotates the inertial coordinate system by an infinitesimal angular displacement, and a boost component which boosts it by an infinitesimal relative velocity (or rapidity).

Also, I have noticed that rapidity plays a very similar role (in the context of Lorentz transformations) to angular position. When purely spatial Lorentz transformations are expressed in terms of angle, the trigonometric cosine and sine are used, whereas when space-time Lorentz boosts are expressed in terms of rapidity, the hyperbolic cosine and sine are used.

Moreover, I have noticed that in Noether's theorem, infinitesimal rotations are generated by the angular momentum pseudovector

I know that for any two four vectors (a

Thus, the quantities

Now, I know the infinitesimal angular displacement d

Since the gradient operator

Thus, the bottom line of all of this is that the angular displacement d

Based on the symmetry between angular displacement and rapidity, and between angular momentum

This would certainly be a nice symmetry between the quantities. I am not sure whether all this reasoning is correct or not. I have tried to research the answer to this online, but nowhere I have seen is this relationship between angular displacement, rapidity, and angular momentum explored to this extent. So I wanted to ask if my reasoning (and my conjecture) is correct or not.

I know that a general infinitesimal Lorentz transformation has two components, a rotation component which rotates the inertial coordinate system by an infinitesimal angular displacement, and a boost component which boosts it by an infinitesimal relative velocity (or rapidity).

Also, I have noticed that rapidity plays a very similar role (in the context of Lorentz transformations) to angular position. When purely spatial Lorentz transformations are expressed in terms of angle, the trigonometric cosine and sine are used, whereas when space-time Lorentz boosts are expressed in terms of rapidity, the hyperbolic cosine and sine are used.

Moreover, I have noticed that in Noether's theorem, infinitesimal rotations are generated by the angular momentum pseudovector

**L**=**x**×**p**, while infinitesimal Lorentz boosts are generated by the polar vector**N**= t**p**- E**x**(where E is the energy).I know that for any two four vectors (a

_{t},**a**_{x}) and (b_{t},**b**_{x}), the pseudovector**a**_{x}×**b**_{x}and the vector a_{t}**b**_{x}-b_{t}**a**_{x}form an antisymmetric relativistic tensor.Thus, the quantities

**L**and**N**form an antisymmetric four-tensor. Similarly, the magnetic and electric fields**B**and**E**form an antisymmetric four-tensor.Now, I know the infinitesimal angular displacement d

**θ**(i.e. rotation) of an inertial frame is a pseudovector that can be expressed as ½(**∇**×δ**x**) (where δ**x**is the vector field that describes the infinitesimal linear displacement at each space-time point attached to the reference frame during an infinitesimal rotation), while the infinitesimal change in rapidity d**φ**describing an infinitesimal Lorentz boost is a polar vector.Since the gradient operator

**∇**acts as the spatial component of a four-vector (-∂/∂t,**∇**) and δ**x**is of course the spatial component of the four-vector (δt,δ**x**), where δt is the scalar field representing the infinitesimal change in local time at each point attached to the reference frame during an infinitesimal Lorentz boost. (Note: Because of the relativity of simultaneity, I believe δt is*not*the same throughout the frame: it varies with space, and thus has a nonzero gradient**∇**(δt))Thus, the bottom line of all of this is that the angular displacement d

**θ**together with the quantity -½ ( ∂/∂t(δ**x**) +**∇**(δt) ) should form an antisymmetric four-tensor.Based on the symmetry between angular displacement and rapidity, and between angular momentum

**L**and rotations on the one hand, and the quantity**N**and Lorentz boosts on the other hand, I have reason to suspect that the quantity -½( ∂/∂t(δ**x**) +**∇**(δt) ) associated with a Lorentz boost is in fact equal to the infinitesimal change in rapidity d**φ**: In other words, my conjecture is that the quantities d**θ**and d**φ**describing an infinitesimal Lorentz transformation form an antisymmetric relativistic tensor, just like**L**and**N**, or like**B**and**E**.This would certainly be a nice symmetry between the quantities. I am not sure whether all this reasoning is correct or not. I have tried to research the answer to this online, but nowhere I have seen is this relationship between angular displacement, rapidity, and angular momentum explored to this extent. So I wanted to ask if my reasoning (and my conjecture) is correct or not.

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