Exploring the Concept of Center of Mass-Energy

[Himanshu]

I have learned that the concept of Center of Mass is universal and is applicable everywhere. As per Einstien's Mass-Energy Equivalence there should be a concept of "center of energy" also? For example if I annihilate electron and positron center of mass-energy should apply. The position of center of mass-energy should not be displaced as no external force is acting on the system.

So does there a concept of "center of mass-energy" exists in theories?

 

[Dale]

Yes, in special relativity it is expressed in the conservation of 4-momentum. Look at the section entitled Conservation of four-momentum at http://en.wikipedia.org/wiki/Four-momentum.

 

[Himanshu]

Yeah! I heard that term before. But I am not clear.

In the wiki-article it says that four-momentum comparise of an energy term and 3 momentum terms. How is it true? How could such a physical quantity exist with two entirely different objects? Or is it just a new formalism?

 

[Hootenanny]

In the wiki-article it says that four-momentum comparise of an energy term and 3 momentum terms. How is it true? How could such a physical quantity exist with two entirely different objects? Or is it just a new formalism?

I'm afraid I don't understand your question, could you elaborate please.

 

[Dale]

Yeah! I heard that term before. But I am not clear.

In the wiki-article it says that four-momentum comparise of an energy term and 3 momentum terms. How is it true? How could such a physical quantity exist with two entirely different objects? Or is it just a new formalism?

That is correct, there is an energy term and 3 momentum terms. One of the things that relativity did is to show the connection between space and time. There is not a separate space and time, but rather a single spacetime that is split into space and time components differently in different reference frames. A direct consequence of this is the fact that energy and momentum are also part of a single entity called the 4-momentum that is split into the timelike energy component and the spacelike momentum components in different ways by different reference frames. This unification of energy and momentum is really the basis of the famous e=mc^2 equation.

 

[Himanshu]

This unification of energy and momentum is really the basis of the famous e=mc^2 equation.

OK! That's where E=MC2 comes from! So E=MC2 didn't come to Einstein in a dream. GREAT! Now I've found it. Thank You.

Can you provide me a link for it that derives the equation mathematically.

 

[belliott4488]

Yeah! I heard that term before. But I am not clear.

In the wiki-article it says that four-momentum comparise of an energy term and 3 momentum terms. How is it true? How could such a physical quantity exist with two entirely different objects? Or is it just a new formalism?

One thing that might be throwing you off a bit is the apparent mixing of units into one vector, i.e. units of momentum in three components and units of energy in the fourth. The thing to keep in mind is that when people talk about four-vectors, they often neglect to mention factors of c that are needed to get the units right (as well as to make the Physics correct), sometimes choosing a system of units where c=1 to make it numerically sensible to leave out.

When you talk about 3 space and 1 time dimension, the fourth dimension should be understood as c*t, which gives you units of length to match the first three dimensions. When you talk of the four-momentum vector, you should think of the component in the fourth dimension as the total energy divided by c, which gives you units of momentum. (See http://en.wikipedia.org/wiki/Four-momentum#Minkowski_norm:_p2" in particular.)

 

[Dale]

Can you provide me a link for it that derives the equation mathematically.

There are many derivations available. I like this approach as it is motivated by the 4-vector formulation:

http://en.wikipedia.org/wiki/Four-vector