Understanding E=mc² and Other Equations of Relativity

  • Thread starter Antuanne
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In summary, the equation E2=m2c4+p2c2 includes relativistic mass. If you substitute the expression for relativistic 3-momentum (p=γmv) into E2=m2c4+p2c2, you'll end up at E=γmc2.
  • #1
Antuanne
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I've just saw this other equation E2=m2c4+p2c2 but what's going on with that? I thought E=mγc2 was the equation including relativistic mass. What is going on?
 
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  • #2
Just algebra. If you substitute the expression for relativistic 3-momentum (p=γmv) into E2=m2c4+p2c2, you'll end up at E=γmc2.
 
  • #3
I get it now, but, if you we're traveling near the speed of light and needed the Lorentz factor, would you make it E2=(mγ)2c4+(mvγ)c2 since you need to put in for relativistic mass in both places?
 
  • #4
Antuanne said:
I get it now, but, if you we're traveling near the speed of light and needed the Lorentz factor, would you make it E2=(mγ)2c4+(mvγ)c2 since you need to put in for relativistic mass in both places?

No, in the formula E2=m^2c^4+p^2c^2, m is rest mass (or invariant mass). In substituting for momentum, it is better to just treat p=γmv as a definition of momentum, not as 'relativistic mass' times velocity.
 
  • #5
Antuanne said:
I get it now, but, if you we're traveling near the speed of light and needed the Lorentz factor, would you make it E2=(mγ)2c4+(mvγ)c2 since you need to put in for relativistic mass in both places?

No. The m in E2=(mc2)2 + p2c2 is the rest mass.

One of the advantages of this formulation is that it works for photons and other massless particles as well.
 
  • #6
That makes sense, but in the simplified thing E=mγc2, where does the γ come from then?
 
  • #7
Antuanne said:
That makes sense, but in the simplified thing E=mγc2, where does the γ come from then?

mc^2 is rest energy. mγc^2 is total energy, including kinetic energy. Not quite sure what you are asking. In Newtonian mechanics, do you ask: where does the 1/2 come from in:

KE = (1/2) mv^2 ?
 
  • #8
What I'm asking is what is the equation E2=m2c4+p2c2 used for if E=mγc2 is used for total energy? And is the γ in the just there for relativistic mass or what is that there for?
 
  • #9
Antuanne said:
That makes sense, but in the simplified thing E=mγc2, where does the γ come from then?
It is always there. But if you are at rest, the gamma factor is 1 and it reduces to the famous form. If you are traveling at lightspeed then gamma is infinite, but m is zero and you need to use the other form and a different definition of momentum - p=2pi hk.
 
  • #10
Ibix said:
It is always there. But if you are at rest, the gamma factor is 1 and it reduces to the famous form. If you are traveling at lightspeed then gamma is infinite, but m is zero and you need to use the other form and a different definition of momentum - p=2pi hk.

Normally, for massless particles traveling at c, you simply use p=E/c.
 
  • #11
Antuanne said:
What I'm asking is what is the equation E2=m2c4+p2c2 used for if E=mγc2 is used for total energy? And is the γ in the just there for relativistic mass or what is that there for?

The two equations are equivalent for particles with non-zero rest mass. Use whichever you like the look of better. However, the longer form is also useful for massless particles, albeit with a different expression for momentum.

The gamma isn't "for" anything. Some people do like to define a "relativistic mass" and make the relativistic equations look like the Newtonian ones. Consensus around here is that this is not helpful - for example you need two different relativistic masses to deal with force, and the acceleration is not parallel to the force (in general) anyway. Better to acknowledge that from the beginning.

The gamma is a consequence of being in a universe with a Minkowski space-time. It is not a modification to Newton. If you wish to recover the low velocity limit, do a binomial expansion of gamma and neglect terms of order v4 and higher. But you can't get from Newton to Einstein that way.

Does that make some kind of sense?
 
  • #12
Do E2=m2c4+p2c2 and E=mγc2 give you the same answer, because, it doesn't seem like they do?
 
  • #13
Antuanne said:
Do E2=m2c4+p2c2 and E=mγc2 give you the same answer, because, it doesn't seem like they do?

Of course they do. Just plug p=mγv in and you get E=mγc2 after algebra.
 
  • #14
Antuanne said:
Do E2=m2c4+p2c2 and E=mγc2 give you the same answer, because, it doesn't seem like they do?

They do. For p2, just substitute m2v2γ2 = m2c4(v/c)2γ2
 
  • #15
But E2=m2c4-p2c2 has v velocity in p so where does that go?
 
Last edited:
  • #16
Did you try the algebra?

Here are a few stages along the way. Try to fill in the gaps.

E^2 = (mc^2)^2 + m^2γ^2v^2c^2

= (mc^2)^2 ( 1 + v^2/(c^2 - v^2) )

= (mc^2)^2 (c^2 / (c^2 - v^2) )

then E = mγc^2
 
  • #17
I still can't figure out how to factor or whatever your doing to get to that!
 
  • #18
Antuanne said:
I still can't figure out how to factor or whatever your doing to get to that!

Don't take this wrong, but while SR needs no higher math, it needs basic facility with algebra. It would be much better for you to review algebra and work it out for yourself than for me fill in one or two more steps between each posted intermediate result.
 
  • #19
Antuanne, don't forget that[tex]\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}[/tex]You will need to use that at some point.
 
  • #20
The OP won't be coming back.
 

1. What is the significance of E=mc² in the theory of relativity?

The equation E=mc², also known as the mass-energy equivalence equation, is a fundamental concept in the theory of relativity. It shows that mass and energy are two sides of the same coin, and that they can be converted into one another. This equation helps us understand the relationship between mass and energy and how they are intertwined in the fabric of the universe.

2. Can you explain the meaning of each term in the equation E=mc²?

The letter E stands for energy, m stands for mass, and c represents the speed of light. The equation essentially states that the energy of an object is equal to its mass multiplied by the speed of light squared. This means that a small amount of mass can contain a large amount of energy, as long as it is moving at the speed of light.

3. How does E=mc² relate to Einstein's theory of relativity?

E=mc² is a key component of Einstein's theory of special relativity, which states that the laws of physics are the same for all observers in uniform motion. This equation shows that mass and energy are not separate entities but are interconnected, and that they both play a role in the fabric of space and time.

4. Can you provide an example of how E=mc² is applied in real life?

One example of the application of E=mc² is in nuclear energy. In a nuclear reaction, a small amount of mass is converted into a large amount of energy, as shown by this equation. This is the principle behind nuclear power plants and nuclear weapons.

5. Are there any other equations of relativity that are important to understand?

Yes, there are several other equations of relativity that are crucial to understanding the theory. Some examples include the Lorentz transformation equations, which describe how the measurements of space and time change for different observers, and the equation for gravitational force, which explains how mass and gravity are related.

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