Derivation of the conservation of total energy and momentum

In summary, Gauss's theorem states that if a divergence exists between two fields in a space, then the fields must go to zero at the boundary.
  • #1
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I want to derive from ##T^{\mu \nu}_{,\nu}=0## the equation: ##\int T_{0\mu}d^3 y=constant##, I don't see how exactly.

From the derivative I know that ##T^{0\mu}_{,\mu}=0##, but I don't see how to integrate this equation, it's ##T^{00}_0+T^{0i}_i=0##.
But how to proceed from here?

Thanks in advance, great forums!
 
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  • #2
Gauss’ theorem.
 
  • #3
$$T^{\mu \nu}{}_{, \nu} = 0 \Leftrightarrow \frac{\partial T^{\mu 0}}{\partial t} = - \frac{\partial T^{\mu i}}{\partial x^i} \\ \int \frac{\partial T^{\mu 0}}{\partial t} d^3 x = - \int \frac{\partial T^{\mu i}}{\partial x^i}d^3 x$$ Right hand side is a divergence. Integrate over all space to get ##\int \frac{\partial T^{\mu 0}}{\partial t} d^3 x = - \int T^{\mu i}## over the boundary at infinity. Demand that the fields go to zero there ##\Longrightarrow \int T^{\mu 0} d^3 x## is conserved.
 
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  • #4
kent davidge said:
$$T^{\mu \nu}{}_{, \nu} = 0 \Leftrightarrow \frac{\partial T^{\mu 0}}{\partial t} = - \frac{\partial T^{\mu i}}{\partial x^i} \\ \int \frac{\partial T^{\mu 0}}{\partial t} d^3 x = - \int \frac{\partial T^{\mu i}}{\partial x^i}d^3 x$$ Right hand side is a divergence. Integrate over all space to get ##\int \frac{\partial T^{\mu 0}}{\partial t} d^3 x = - \int T^{\mu i}## over the boundary at infinity. Demand that the fields go to zero there ##\Longrightarrow \int T^{\mu 0} d^3 x## is conserved.
This is fine, but obscures the beauty of using Gauss’ theorem in 4 dimensions.
 
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  • #5
kent davidge said:
$$T^{\mu \nu}{}_{, \nu} = 0 \Leftrightarrow \frac{\partial T^{\mu 0}}{\partial t} = - \frac{\partial T^{\mu i}}{\partial x^i} \\ \int \frac{\partial T^{\mu 0}}{\partial t} d^3 x = - \int \frac{\partial T^{\mu i}}{\partial x^i}d^3 x$$ Right hand side is a divergence. Integrate over all space to get ##\int \frac{\partial T^{\mu 0}}{\partial t} d^3 x = - \int T^{\mu i}## over the boundary at infinity. Demand that the fields go to zero there ##\Longrightarrow \int T^{\mu 0} d^3 x## is conserved.
kent davidge said:
$$T^{\mu \nu}{}_{, \nu} = 0 \Leftrightarrow \frac{\partial T^{\mu 0}}{\partial t} = - \frac{\partial T^{\mu i}}{\partial x^i} \\ \int \frac{\partial T^{\mu 0}}{\partial t} d^3 x = - \int \frac{\partial T^{\mu i}}{\partial x^i}d^3 x$$ Right hand side is a divergence. Integrate over all space to get ##\int \frac{\partial T^{\mu 0}}{\partial t} d^3 x = - \int T^{\mu i}## over the boundary at infinity. Demand that the fields go to zero there ##\Longrightarrow \int T^{\mu 0} d^3 x## is conserved.
Thanks for your illuminating answer!
 
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  • #6
The important point to remember is that only if ##\partial_{\nu} T^{\mu \nu}=0##, the naive calculation of the total four-momentum density leads to a four-vector,
$$g^{\mu}=\int_{mathbb{R}^3} \mathrm{d}^3 \vec{x} T^{\mu 0}.$$
The proof follows again from Gauss's theorem in four dimensions. For the proof (for the completely analogous case of a conserved four-current) see p. 19 in

https://itp.uni-frankfurt.de/~hees/publ/kolkata.pdf
 

Related to Derivation of the conservation of total energy and momentum

1. What is the conservation of total energy and momentum?

The conservation of total energy and momentum is a fundamental principle in physics that states that the total amount of energy and momentum in a closed system remains constant over time. This means that energy and momentum can be transferred or transformed, but the total amount in the system will always remain the same.

2. How is the conservation of total energy and momentum derived?

The conservation of total energy and momentum can be derived from the laws of physics, specifically the laws of conservation of energy and momentum. These laws state that energy and momentum cannot be created or destroyed, only transferred or transformed. By applying these laws to a closed system, we can show that the total energy and momentum must remain constant.

3. What are some real-life examples of the conservation of total energy and momentum?

One example of the conservation of total energy and momentum is a billiard ball game. When a ball is hit with a cue stick, the energy and momentum of the cue stick is transferred to the ball, causing it to move. However, the total amount of energy and momentum in the system (the cue stick and the ball) remains the same.

4. Why is the conservation of total energy and momentum important?

The conservation of total energy and momentum is important because it is a fundamental principle that governs the behavior of physical systems. It allows us to make predictions and calculations about the behavior of objects and systems, and it is a key concept in understanding the laws of motion and energy conservation.

5. Are there any exceptions to the conservation of total energy and momentum?

In most cases, the conservation of total energy and momentum holds true. However, there are some exceptions, such as in nuclear reactions or when dealing with very small particles at the quantum level. In these cases, energy and momentum can be created or destroyed, but the total amount is still conserved through other means, such as the conversion of mass into energy.

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