Kinetic energy distribution in high speeds and large scales

[RyanH42]

We know that If object is homogenius and small (like gold atom) we can use this equation to calculate total energy of object when it moves very slow due to speed of light.And the equation is $1/2(m_1v^2+m_2v^2+m_3v^2...=1/2Mv^2$, $(M=m_1+m_2+m_3...)$ here v is small and we are talking about a sphere.This means in every piece there's extra kinetic energy.And when we add them we get total energy

Now let's suppose we have a very large scale object, a galaxy cluster.Its not homogenius.So let's suppose that massive bigger object moves nearly speed of light.How can we calculate the the kinetic energy distrubition ?

Is this kinetic energy will distrubute the sphere in homogenius ways or just where the matter exist ?

If we take larger sphere radius How can this kinetic energy distrubition can be change ?

 

[mfb]

A gold atom is not homogeneous. It does not matter, however.

This means in every piece there's extra kinetic energy.

Depends on the way you count contributions to the total energy.

For the total kinetic energy, you can always consider the system in its rest frame first - calculate the total mass of the system there (energy divided by speed of light squared). Then calculate the total energy based on this mass and its speed. Both can be done with the usual relativistic formulas.
For the kinetic energy of individual components, just do the same thing as above but restricted to this component.

 

[RyanH42]

So exaple we have extra 5mc^2 kinetic energy.Now our sphere volume is 100 m^3 and in the sphere there 10 objects.Is this kinetic energy can spread the sphere in homogenius ways.I mean can we say in every 1m^3 there's 5mc^2/100m^3 energy ? Or we need to say this kinetic energy inside the mass it gives extra mass to the rest mass ?

 

[mfb]

.Is this kinetic energy can spread the sphere in homogenius ways.I mean can we say in every 1m^3 there's 5mc^2/100m^3 energy ?

That depends on the details of your object. If it is completely homogeneous, that is a meaningful statement.

 

[RyanH42]

Its not completely homogenius.So this time my other statemnet will true ?Or there's a theory which describes this non-homogenius situation ?

 

[mfb]

You can always calculate the energy of arbitrary smaller volumes in your system, in the same way you can do it for the whole system.

 

[RyanH42]

If I can calculate always smaller arbitary numbers can I get a equation ?Or a source ? Or can you describe me using my example ?

 

[mfb]

I think post 2 has a full description.

 

[RyanH42]

Then thank you.