Implications of the Mass-energy equivalence

[jpo]

Hello All,

Let m be a mass, equivalent to energy E such, that E=mc$^{2}$.
Does it follow that c is the cosmic speed limit?
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To say the above with more words:
1) m is a mass
2) in some process, it is established that through removal/vanishing/anihilation/etc of m, energy E is released
3) it is established that E can be expressed as E= mc$^{2}$.

What are the implications of such energy-mass relationship? Does it follow that c must be the cosmic speed limit? What other, no matter how insignificant, implications does E= mc$^{2}$ have on the parameter c?

Regards

 

[jpo]

I guess I am trying to reason it backwards -

commonly, assuming a cosmic speed limit leads to special relativity relationships (length, time interval etc); conservation of the 4-dimensional momentum leads to E=mc$^{2}$

Can the relativity construct be built if we start at E=mc$^{2}$ and follow backwards through the relativity argument.

Suppose someone wrote E=mc$^{2}$, not knowing anything about relativity or the meaning of c. In that case, if E=mc$^{2}$ is accepted as true, what follows from it?

 

[Nugatory]

What are the implications of such energy-mass relationship? Does it follow that c must be the cosmic speed limit? What other, no matter how insignificant, implications does E= mc$^{2}$ have on the parameter c?

As you say, E= mc$^{2}$ is a conclusion not a starting point.

If you start with that conclusion, interpret it as suggesting that an object gains mass as its kinetic energy increases, and then work backwards, you can show that there is no necessary conflict with the two postulates of special relativity, nor with conservation of energy and momentum. But all you've really done in that exercise is demonstrate that special relativity is self-consistent.

You're asking "If B (E= mc$^{2}$) follows from A (the postulates of SR), then what can I say about A if I start by assuming that B is true?". The answer is "If B is true, then maybe A is true" and from there you can get to "maybe everything else that follows from A is also true". But you won't get beyond that point unless you can show that not only does A imply B, but also B implies A.

 

[elfmotat]

Well, it follows that c would need to be a universal constant. If you then take the general transformation between inertial frames:

$$\begin{bmatrix} t' \\ x' \end{bmatrix} = \frac{1}{\sqrt{1+\kappa v^2}} \begin{bmatrix} 1 & -\kappa v \\ -v & 1 \end{bmatrix} \begin{bmatrix} t \\ x \end{bmatrix}$$

where $\kappa$ is some universal constant with dimensions 1/v². One might therefore suspect that $\kappa =-1/c^2$.

From this transformation law (the Lorentz transformation), it follows that c is also a universal speed limit.

 

[jpo]

Nugatory,

If you start with E= mc$^{2}$, interpret it as suggesting that an object gains mass as its kinetic energy increases, ...

No motion is implied by E= mc$^{2}$, am I wrong? E appears as a result of converting REST MASS m to E.

How knowing E= mc$^{2}$ would prompt us to look at moving mass m?

 

[jpo]

elfmotat,

Well, it follows that c would need to be a universal constant.

This sounds interesting... but HOW does it follow from E=mc$^{2}$?

As for looking at moving frames - the question again is - since E is a result of converting REST MASS, what would prompt us to consider motion, let alone moving frames? If we started looking at moving frames, we will eventually derive SR, but this would not be a logical consequence of E=mc$^{2}$, but sheer luck.

The assumption here is we know E=mc$^{2}$ only and NOTHING about relativity yet. Will E=mc$^{2}$ logically lead us to relativity and the concept c=const (cosmic speed limit)

 

[sweet springs]

hi.

E^2 - p^2 = m^2 proper constant in any coordinates is essence. Here I put c=1 nor E^2 - p^2 c^2 = m^2 c^4.
regards.

 

[elfmotat]

elfmotat,

This sounds interesting... but HOW does it follow from E=mc$^{2}$?

Well, sort of by definition. If c² is a universal conversion factor between E and m, then it's by definition a universal constant.

 

[Nugatory]

No motion is implied by E= mc$^{2}$, am I wrong? E appears as a result of converting REST MASS m to E.

How knowing E= mc$^{2}$ would prompt us to look at moving mass m?

If the energy E increases with increasing velocity (it does - that's what kinetic energy is all about) and the equality holds, then the right-hand side has to increase too.