Question

Prove the 'zero component' lemma for an antisymmetric tensor \(T^{\mu v}\) : if any one of its off-diagonal components is zero in all inertial frames, then the entire tensor is zero.

Short answer

#Answer# Since one off-diagonal component is zero in all inertial frames, it implies that all the other components of the antisymmetric tensor are zero as well, proving the 'zero component' lemma for an antisymmetric tensor.

Step-by-step solution

Write down the properties of an antisymmetric tensor

An antisymmetric tensor \(T^{\mu v}\) has the property that when the indices are swapped, the tensor changes its sign, i.e. \(T^{\mu v} = -T^{v\mu}\).

Consider an off-diagonal component being zero

Let's assume that one off-diagonal component of the tensor is zero in all inertial frames. Without loss of generality, let's assume that this component is when \(\mu=1\) and \(v=2\), or \(T^{12}=0\) in all inertial frames. We will use this assumption to show that all the other components must also be zero.

Use the antisymmetric property for the non-diagonal components

If the off-diagonal component \(T^{12}\) is zero, then using the antisymmetric property, we get: \(T^{21} = -T^{12} = -0 = 0\) So, both \(T^{12}\) and \(T^{21}\) are zero. Now, let's consider the other possible pairs of indices, \((1,0)\), \((1,3)\), \((2,0)\), \((2,3)\), and \((3,0)\). For each pair, we use the antisymmetric property to show that the corresponding component is zero: - For \(T^{10}\), we have \(T^{01} = -T^{10}\). Since \(T^{12}=0\), we know that the off-diagonal component \(T^{01}\) must also be zero in all inertial frames. Therefore, \(T^{10} = -T^{01} = 0\). - Similarly, for \(T^{13}\), we have \(T^{31} = -T^{13}\) and as \(T^{12}=0\), the off-diagonal component \(T^{31}\) must also be zero in all inertial frames. Therefore, \(T^{13} = -T^{31} = 0\). - Continuing this logic for the other components, we get: - \(T^{20} =-T^{02} = 0\) - \(T^{23} = -T^{32} = 0\) - \(T^{30} = -T^{03} = 0\)

Conclude that all off-diagonal components are zero

Since all the off-diagonal components are zero, the entire tensor \(T^{\mu v}\) is antisymmetric with zero off-diagonal components. This means that \(T^{\mu v}\) is zero for all indices \(\mu\) and \(v\), proving the 'zero component' lemma for an antisymmetric tensor.

Key concepts

Zero Component Lemma

The "zero component lemma" for antisymmetric tensors is a fascinating concept in mathematics and physics. It asserts that if any one of the off-diagonal components of an antisymmetric tensor is zero in all inertial frames, then the entire tensor must be zero. Here's why it's true: Antisymmetric tensors have a unique property: they change sign when their indices are swapped. This means that if you have a tensor component like \( T^{\mu v} \), and you switch the indices to \( T^{v\mu} \), the value of the tensor becomes \(-T^{\mu v}\). Suppose we take an example where \( T^{12} = 0 \). Due to the antisymmetric nature, \( T^{21} = -T^{12} = -0 = 0 \). This zero property propagates to all components, as swapping indices maintains zero-ness due to antisymmetry, and due to the exhaustive checking in inertial frames, proving the lemma.

Inertial Frames

In understanding the zero component lemma, it's essential to grasp the concept of inertial frames. An inertial frame is a reference frame in which objects that are free from external forces move in straight lines at a constant speed. In physics, solving problems within these frames simplifies laws of motion since no fictitious forces like Coriolis force or centrifugal force need to be considered. When we claim that a tensor component is zero in all inertial frames, we're asserting that it remains zero regardless of the observer's motion, provided the observer remains in an inertial frame. This universality adds robustness to the zero component lemma, as it proves the tensor's non-existence consistently across all observers not under acceleration.

Tensor Properties

Tensors are quite powerful in physics since they provide a way to describe physical properties irrespective of coordinate systems. An antisymmetric tensor, like the one we're discussing, has specific attributes: swapping its indices results in the tensor negating itself, i.e., \( T^{\mu v} = -T^{v\mu} \). Also, because of their structure, the diagonal components (like \( T^{00} \), \( T^{11} \), ...) of an antisymmetric tensor are inherently zero, as no number equals its negative unless it is zero: \( T^{\mu \mu} = -T^{\mu \mu} \Rightarrow 2T^{\mu \mu} = 0 \Rightarrow T^{\mu \mu} = 0 \).

• Antisymmetric Tensor: Sign flips when indices do.
• Zero Diagonals: The diagonal elements are always zero.
• Versatility: Useful in describing physical situations independently of observer's velocity or position, especially in relativity.

These tensors serve as essential tools in understanding electromagnetism, angular momentum, and even general relativity, making understanding their properties key to more advanced physics.