Density change of an object with mass

[sudipmaity]

Homework Statement

This is an introductory special theory of relativity short question from university question paper that carried 2marks.
Does the density of an object change as its speed increases? if yes by what factor.

Homework Equations

m=m₀/√[1- (v/c)²]
L=L₀√[1-(v/c)²]

The Attempt at a Solution

My guess is yes .
If 1/√[1-(v/c)²]=γ lorentz factor
ρ=m/L³=m/volume(V)=(m⁰/V₀) *γ⁴
ρ/ρ₀=γ⁴.
No textbook has exclusively covered this problem except for one.
There it is written for proper mass density
ρ/ρ₀=γ and it is γ² for mass-energy density.
And V=V₀/γ.
The author only mentioned the relations.Did not derive them either concisely or in detail.
I am now confused.How come lorentz factor has the same power for both length and volume?
Why and how does proper mass density and mass-energy density have different factor?

 

[phyzguy]

If an object has a velocity in the X-direction, are all three spatial dimensions Lorentz contracted?

 

[sudipmaity]

The answer to your question is no.But nobody said here about a velocity in x direction.What if for the most general case the object is traveling in an 3d space.Shouldn't all the components contract?

 

[phyzguy]

No, all components don't contract, only the component in the direction of motion. It doesn't matter what direction it is, I just used the X-direction to make the question simpler. If it is moving with a uniform velocity, I can always rotate my coordinate system and call the direction of motion the X-direction.

 

[sudipmaity]

Ok so that clears up the volume factor. Could you please explain the proper mass density and mass energy density factor difference?

 

[sudipmaity]

After thinking on my own question what I feel is in the case PROPER MASS DENSITY variation is only due to contraction in dimension of the body parallel to its direction of motion.Hence V=V₀/γ.
m=m₀.
So ρ=ρ₀γ.
But for MASS-ENERGY DENSITY both the mass and volume change.
Mass increases :m=m₀γ
Volume changes as above.
So ρ=ρ₀γ².
Am I right?
Do you guys concur?
If yes then what should be right answer to my first post?
Should it be γ or γ²?

 

[phyzguy]

Your reasoning is correct. As to the answer to your first question, it depends on which they are asking about. I would think that the word density refers to the proper density, but I'm not really sure.

 

[sudipmaity]

What still makes my mind itch is my inability to explain the answer physically.
My answer was clearly out of mathematical intuition.
Can somebody explain PHYSICALLY why proper mass density varies by a factor γ while mass energy density by γ²?
What could be the best possible QUALITATIVE explanation?