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Prove that if \(T\) and \(a\) are respectively a four-tensor and a four-vector, then \(b=T \cdot a=T G a\) is a four-vector; that is, it transforms according to the rule \(b^{\prime}=\Lambda b\)

Short Answer

Expert verified
Yes, the product is a four-vector; it obeys the transformation rule.

Step by step solution

01

Understanding the Problem

We need to prove that the product of a four-tensor \(T\) and a four-vector \(a\) results in a four-vector \(b = T \cdot a\). We also need to show that \(b\) transforms according to the four-vector transformation rule, \(b^{\prime} = \Lambda b\). Here, \(\Lambda\) is the Lorentz transformation matrix.
02

Defining the Four-Tensor and Four-Vector

A four-vector \(a\) transforms under Lorentz transformation \(\Lambda\) as \(a' = \Lambda a\). A rank-two four-tensor \(T\) transforms as \(T' = \Lambda T \Lambda^T\), where \(\Lambda^T\) is the transpose of \(\Lambda\).
03

Expressing the Product

The product \(b = T \cdot a\) is given by a tensor-vector multiplication. The elements of \(b\) are expressed as \(b^\mu = T^{\muu} a_u\), where the Einstein summation convention is implied over the index \(u\).
04

Applying Lorentz Transformation to the Product

Under Lorentz transformation, the product becomes \(b'^\mu = T'^\mu_{\,\sigma} a'^\sigma\). Substituting the transformation properties, we have \(b'^\mu = \Lambda^\mu_{\,\rho} T^{\rho\tau} \Lambda^\tau_{\,u} a^u\).
05

Simplifying the Transformation Expression

Using associativity, the expression simplifies to \(b'^\mu = \Lambda^\mu_{\,\rho} (T^{\rho\tau} a_\tau) = \Lambda^\mu_{\,\rho} b^\rho\). This matches the transformation rule for four-vectors, \(b'^\mu = \Lambda^\mu_{\,\rho} b^\rho\), confirming that \(b\) is a four-vector.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Four-tensor
In the realm of relativity, four-tensors extend the concept of tensors into four dimensions. They allow us to understand how physical quantities transform under the changes of reference frames, a key aspect of Einstein's theory of relativity.
Four-tensors can have various ranks, which denote their array structure and complexity.
Commonly discussed is the rank-two four-tensor. It has components represented by two indices, like in the exercise where tensor transposes as \( T' = \Lambda T \Lambda^T \).
  • \( \Lambda \) is the Lorentz transformation matrix.
  • \( \Lambda^T \) indicates its transpose.
Rank-two tensors are useful to represent measurable quantities like the electromagnetic field tensor. They differ from lower-rank tensors in terms of complexity and their interactions with other physical entities.
Four-vector
Four-vectors are vital constructs in the study of relativity. They combine both spatial and temporal components into a single mathematical entity. These vectors are essential as they make the mathematical representation of physical laws invariant under Lorentz transformations.
A four-vector's transformation under a Lorentz transformation \(\Lambda\) is given by:
  • \( a' = \Lambda a \)
This expression shows how the components of a four-vector combine space and time, ensuring that the laws of physics hold true regardless of the observer's velocity. An example of a four-vector is the four-momentum, which combines the relativistic energy and momentum of a particle. They reflect the isotropic nature of the spacetime continuum. Any object expressed as a result, like vector \( b \) in the exercise, inherently follows the transformation rule to remain consistent in different inertial frames.
Einstein summation convention
The Einstein summation convention streamlines expressions involving tensor calculus by implying a sum over indices appearing twice in a term. Instead of explicitly writing out the sum, the repeated index suggests a summation, making equations less cumbersome and easier to manage.
In our exercise, when expressing the product \( b = T \cdot a \), the components \( b^\mu = T^{\mu u} a_u \), automatically sum over the repeated index \( u \).
  • This eliminates the need for the cumbersome \( \sum \), implied by the repetition of index.
  • Speeds up calculations in complex tensor equations.
This notation is particularly powerful in relativity and field theory, where tensors represent diverse physical quantities. The ability to simplify tensor equations is one of the reasons for its widespread use in advanced physics and mathematics.

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Most popular questions from this chapter

A mad physicist claims to have observed the decay of a particle of mass \(M\) into two identical particles of mass \(m,\) with \(M<2 m .\) In response to the objections that this violates conservation of energy, he replies that if \(M\) was traveling fast enough it could easily have energy greater than \(2 m c^{2}\) and hence could decay into the two particles of mass \(m\). Show that he is wrong. [He has forgotten that both energy and momentum are conserved. You can analyse this problem in terms of these two conservation laws, but it is much simpler to go to the rest frame of \(M .]\)

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