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Bibipandi
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I'm studying General Relativity and facing several problems. We know that energy-momentum must be Lorentz invariant in locally inertial coordinates. I am not sure I understand this point clearly. What is the physics behind?
Dickfore said:It's not. It's covariant.
Bibipandi said:I'm studying General Relativity and facing several problems. We know that energy-momentum must be Lorentz invariant in locally inertial coordinates. I am not sure I understand this point clearly. What is the physics behind?
kev said:I am assuming that the term on the right is what you are calling the energy-momentum.
kev said:If we take the well known Energy equation:
[tex]E = \sqrt{ (m_oc^2)^2 + p^2c^2} = \sqrt{ (m_oc^2)^2 + \frac{(m_o v c)^2}{(1-v^2/c^2)}} [/tex]
where
E is the total energy [itex] (m_0c^2)/\sqrt{(1-v^2/c^2)}[/itex],
[itex]m_oc^2[/itex] is the rest energy ,
p is the momentum
and re-arrange it so that:
[tex]m_oc^2 = \sqrt{E^2 - p^2c^2} [/tex]
then it is easy to see that the term on the left is invariant because the rest mass and the speed of light are both invariants, so the Energy-Momentum term on the right must be invariant too. I am assuming that the term on the right is what you are calling the energy-momentum.
Bibipandi said:What you are mentioning about is the Energy-Momentum four-vector [tex]p^{\mu} (p^0 =m_0 c^2, p^i)[/tex]. Not surprisingly, the four-vector is Lorentz invariant because one always has: [tex]p^{\mu}p_{\mu}=\sqrt{E^2 - p^2c^2}=m_{0}c^2 [/tex]. The physics behind is the validity of the well-known equation [tex]E=mc^2[/tex] in the rest frame leading to the Lorentz invariance of the four-vector.
What I am talking about is the similar physics behind in the case of Energy-Momentum tensor.
Bibipandi said:What you are mentioning about is the Energy-Momentum four-vector [tex]p^{\mu} (p^0 =m_0 c^2, p^i)[/tex]. Not surprisingly, the four-vector is Lorentz invariant because one always has: [tex]p^{\mu}p_{\mu}=\sqrt{E^2 - p^2c^2}=m_{0}c^2 [/tex].
bcrowell said:No, the energy-momentum four-vector is not Lorentz invariant. It transforms like a four-vector. Its *magnitude* is a Lorentz scalar, so its *magnitude* is invariant.
The subject line of your original post was "Why is energy-momentum tensor Lorentz invariant?" If this refers to the stress-energy tensor, then the answer is that it isn't invariant; it transforms as a rank-2 tensor. If this refers to the energy-momentum four-vector, then the answer is that it isn't invariant; it transforms as a four-vector.
Are you confusing "invariant" with "conserved?"
Bibipandi said:I'm studying General Relativity and facing several problems. We know that energy-momentum must be Lorentz invariant in locally inertial coordinates. I am not sure I understand this point clearly. What is the physics behind?
The energy-momentum tensor is a mathematical object used in physics to describe the distribution of energy and momentum in a given system. It is a rank-2 tensor, meaning it has both covariant and contravariant indices, and can be used to describe the energy and momentum density, as well as the flow of these quantities.
The energy-momentum tensor plays a crucial role in Einstein's general theory of relativity, as it is the source of curvature in spacetime. It also appears in other areas of physics, such as electromagnetism and quantum field theory, and is a fundamental concept in understanding the conservation of energy and momentum in physical systems.
A Lorentz invariant quantity is one that is unaffected by Lorentz transformations, which are mathematical transformations that describe how physical quantities appear to an observer in a different frame of reference. In the context of the energy-momentum tensor, this means that its components will have the same values regardless of the frame of reference in which they are measured.
The principles of relativity state that the laws of physics should be the same for all observers, regardless of their frame of reference. The Lorentz invariance of the energy-momentum tensor ensures that the conservation of energy and momentum, which are fundamental principles in physics, hold true for all observers, regardless of their relative motion.
If the energy-momentum tensor were not Lorentz invariant, it would mean that the conservation of energy and momentum would not hold true for all observers, violating the principles of relativity. This would have significant consequences for our understanding of the laws of physics and could potentially lead to inconsistencies and contradictions in our theories.