Four-Vectors: Definition Issues & Exam Prep

In summary, a four-vector is a set of four numbers that describe an event in Minkowski space, with the first number being the time component and the other three being the spatial components. It differs from a 3-dimensional vector in that it can undergo a Lorentz transformation and remain a four-vector. To show that a vector is a four-vector, one can demonstrate its transformation properties or show that it contracts with another four-vector to form a Lorentz-invariant scalar.
  • #1
Bigfoots mum
14
0
Im having some four-vector definition issues. I have a relativity exam coming up and they quite often ask about 4-vectors.

1) Does this definition sound ok?
'A four-vector is 4 numbers, say X=(X0, X1, X2, X3), used to describe an event in minkowski space. The 'zeroth' is the time component, while the other 3 components are the spatial components of a 3-vector. A four-vector differs from a 3-dimensional vector in that it can undergo a lorentz transformation and remain a four-vector. '
2) How do i show that a vector is actually a four vector?
Do i just show that it remains a valid four vector under a lorentz transformation?

Any help is greatly appreciated
thanks
 
Physics news on Phys.org
  • #2
Hi, the best way to define 4 vectors are by their transformation properties, which is essentially what you have said. In equation form it is

[tex]\widehat{x}^{\mu}=\Lambda^{\mu}_{\nu}x^{\nu}[/tex]

Here Lambda is the mu, nu component of the lorentz matrix, and the einstein summation convention is used.
 
  • #3
Bigfoots mum said:
2) How do i show that a vector is actually a four vector?
Do i just show that it remains a valid four vector under a lorentz transformation?
Yep, that works. I think it's also possible to show that it's a four-vector by demonstrating that if you contract it with another four-vector, the result is Lorentz-invariant (i.e. is a scalar). Sometimes that might be easier.
 

What is a four-vector?

A four-vector is a mathematical representation of a physical quantity in four-dimensional spacetime. It contains four components, typically representing three spatial dimensions and one time dimension, and can be used to describe the position, momentum, or energy of a particle or system.

What are the components of a four-vector?

The components of a four-vector depend on the physical quantity being represented. In general, they will include three spatial components and one time component. For example, a four-vector representing the position of a particle would have components x, y, z, and t, while a four-vector representing the energy of a particle would have components E, px, py, and pz.

How do four-vectors relate to special relativity?

Four-vectors are a key part of the mathematical framework of special relativity. The components of a four-vector must transform in a specific way under Lorentz transformations, which are the transformations that describe the way physical quantities appear to observers in different reference frames moving at constant velocities relative to one another.

What are some common issues with defining four-vectors?

One common issue is the choice of reference frame, as different observers may measure different components for the same four-vector. Another issue is the choice of units, as four-vectors can have different numerical values depending on the units used. Additionally, there may be difficulties in defining four-vectors for certain physical quantities, such as angular momentum.

How can I prepare for an exam on four-vectors?

To prepare for an exam on four-vectors, it is important to have a strong understanding of the mathematical principles and equations involved. Practice problems and sample questions can also be helpful in preparing for the exam. It may also be beneficial to review the connections between four-vectors and other concepts in physics, such as special relativity and quantum mechanics.

Similar threads

  • Advanced Physics Homework Help
Replies
3
Views
799
  • Advanced Physics Homework Help
Replies
1
Views
976
  • Advanced Physics Homework Help
Replies
20
Views
4K
  • Special and General Relativity
Replies
28
Views
2K
Replies
9
Views
1K
Replies
2
Views
1K
Replies
1
Views
578
  • Special and General Relativity
Replies
8
Views
1K
  • Advanced Physics Homework Help
Replies
15
Views
2K
  • Special and General Relativity
Replies
31
Views
8K
Back
Top