# Constructing Lorentz Basis: Restrictions & Possibilities

• parton
In summary, the program reports an infinite solution for relativistic kinetic energy when you enter a speed of light as the velocity.

#### parton

I found an interesting statement in the following exercise 3.2 a), but I'm confused:

Why is there the restriction to $$Y_{0}^{0} > 1$$ and $$det( Y_{\mu}^{\lambda} ) = 1$$ in the definition of a Lorentz basis? In the literature you can also find the definition withouth the restriction to proper Lorentz-Transformations. So my question is if it is always possible to construct a basis with this restrictions and why?

Thanks

Last edited by a moderator:
I think it's based on the following: The set of proper Lorentz transformations have the property $\textrm{det}(\Lambda_\mu^\nu) = 1$ and $\Lambda_0^0$. When acting on the basis states $Y_\mu$ we end up with new basis states. The proper Lorentztransformations leaves the restrictions you mention intact. So if we start out with a Lorentz basis, and we only consider proper Lorentz transformations, then we will always deal with bases that have this property.

Also, I think that with such a choice of bases we also have a "natural" time direction and handedness of the coordinate system. Proper Lorentztransformations leave these properties intact.

I've a question about Relativistic Kinetic Energy. I understand the Proof but I see an infinite solution on the hyperphysics website when I put in the speed of light as the velocity v.

To reproduce

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/releng.html#c6
Enter mass m=1, 10^1
Enter velocity 2.998 10^8 which is C Metres Per Second and in the Javascript as C

I get K.E.(relativistic) Infinite solution.

Shouldn't the solution be E=MC^2? or 10 * (2.998 10^8)^2 where V=C?

Is it a bug in the program or a bug in my reasoning?

Right so, I get it now. We're not really dealing with the Relative Kinetic Energy of the moving object. We dealing with the work done on it. Thanks for your lack of response.

Right so, I get it now. We're not really dealing with the Relative Kinetic Energy of the moving object. We dealing with the work done on it. Thanks for your lack of response.

You probably would have gotten a response
if you started your own thread and
not added to an existing thread that is not directly related to your question.

who said that you could set v=c in that equation for the relativistic kinetic energy?
Mathematically, [with a fixed value for m_0 anc c] you get infinity... as reported by the program.
(The relativistic generalization of the so-called work-energy theorem still applies here:
the [relativistic] net work done on it is equal to the change in its [relativistic] kinetic energy.)

## 1. What is a Lorentz basis?

A Lorentz basis is a set of vectors that form a basis for a particular space, such as the space of 4-vectors in special relativity. These vectors have a special relationship with the Lorentz transformation, which is a mathematical tool used to describe the effects of time and space on objects moving at high speeds.

## 2. Why is constructing a Lorentz basis important?

Constructing a Lorentz basis is important because it allows us to mathematically describe the effects of time and space on objects moving at high speeds. This is essential for understanding and making predictions about phenomena such as time dilation and length contraction in special relativity.

## 3. What are the restrictions in constructing a Lorentz basis?

There are two main restrictions in constructing a Lorentz basis. The first is that the vectors must be linearly independent, meaning that none of them can be written as a linear combination of the others. The second restriction is that the vectors must satisfy the Lorentz transformation equations, which describe how the basis vectors change under the effects of time and space.

## 4. What are the possibilities for constructing a Lorentz basis?

There are many possibilities for constructing a Lorentz basis, as long as the two restrictions mentioned above are satisfied. One common approach is to use the standard basis vectors (1,0,0,0), (0,1,0,0), (0,0,1,0), and (0,0,0,1), and then apply the Lorentz transformation to these vectors to obtain a new set of basis vectors. However, any set of linearly independent vectors that satisfy the Lorentz transformation equations can also be used.

## 5. How does constructing a Lorentz basis relate to other areas of physics?

Constructing a Lorentz basis is closely related to the concept of a reference frame, which is a fundamental concept in both classical and modern physics. It is also essential for understanding the principles of special relativity, including the constancy of the speed of light and the equivalence of space and time. Additionally, the use of Lorentz bases extends to other areas of physics, such as quantum field theory and cosmology.