Find Lorentzian Scalar Product on 4-D Lie Algebra G

In summary, the conversation discusses finding a Lorentzian scalar product that is left invariant on a four-dimensional Lie group with g as its Lie algebra. The person is having trouble finding an equivalent to the classical algebra's scalar product of tr(AB^t). They are wondering if the generator or structure constant must follow a certain criterion in order for this to work. However, there is confusion about the generators and what exactly a left invariant scalar product means, as well as what is meant by Lorentzian in this context.
  • #1
kroni
80
10
Hi everybody,

Let G a four dimmensionnal Lie group with g as lie algebra. Let T1 ... T4 the four generator. I would like to find à lorentzian scalar product (1-3 Signature) on it and left invariant. A classical algebra take tr (AB^t) as scalar product but I don't find à lorentzian équivalent. Did it contraint the generator or the structure constant to respect some criterion ?

Thanks for your answer

Clement
 
  • #3
kroni said:
Hi everybody,

Let G a four dimmensionnal Lie group with g as lie algebra. Let T1 ... T4 the four generator. I would like to find à lorentzian scalar product (1-3 Signature) on it and left invariant. A classical algebra take tr (AB^t) as scalar product but I don't find à lorentzian équivalent. Did it contraint the generator or the structure constant to respect some criterion ?

Thanks for your answer

Clement
I've read your post now several times. I'd like to help you but I don't understand it. The ##T_i## are generators of ##g## or of ##G##? What do you mean by a left invariant scalar product? And Lorentzian in this context means exactly what? Since a four dimensional Lie Algebra isn't semisimple I have difficulties to understand which bilinear form you're looking for.
 

1. What is a Lorentzian Scalar Product on 4-D Lie Algebra G?

A Lorentzian Scalar Product on 4-D Lie Algebra G is a mathematical operation that takes two elements of the 4-D Lie Algebra G and produces a real number. It is a special type of scalar product that is defined in a space-time with a non-Euclidean metric, such as the Minkowski space-time in Einstein's theory of relativity.

2. What is the significance of finding a Lorentzian Scalar Product on 4-D Lie Algebra G?

The significance of finding a Lorentzian Scalar Product on 4-D Lie Algebra G lies in its applications in physics, particularly in theories of relativity. It allows for calculations and measurements to be made in non-Euclidean spaces, which is essential in understanding the behavior of matter and energy in the universe.

3. How is a Lorentzian Scalar Product on 4-D Lie Algebra G calculated?

A Lorentzian Scalar Product on 4-D Lie Algebra G is calculated using the metric tensor, which is a mathematical tool used to define the metric of a space-time. The calculation involves taking the inner product of two elements of the 4-D Lie Algebra G and then applying a transformation based on the metric tensor.

4. Can a Lorentzian Scalar Product on 4-D Lie Algebra G be extended to higher dimensions?

Yes, a Lorentzian Scalar Product on 4-D Lie Algebra G can be extended to higher dimensions. In fact, it is a generalization of the standard Euclidean scalar product, which can be extended to any number of dimensions. However, the calculation becomes more complex as the number of dimensions increases.

5. How is a Lorentzian Scalar Product on 4-D Lie Algebra G used in physics?

A Lorentzian Scalar Product on 4-D Lie Algebra G is used in physics to calculate physical quantities, such as energy and momentum, in non-Euclidean spaces. It is also used in theories of relativity to describe the behavior of matter and energy in space-time. Additionally, it is a fundamental tool in the study of black holes and gravitational waves.

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