Why are we not able to move at the speed of light again?

Question

In many many answers here and papers everywhere, it's often stated that no object can move faster than light.

Why is that again?

Answer 1

For a massive particle to move at, or faster than, the speed of light, it would require infinite energy as shown by Einstein's relativistic equation: $$ E = \gamma \cdot mc^2\quad\left(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\right)\\ E = \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}\\ v\rightarrow c, E \rightarrow \infty $$

If we plug in $c$ for the velocity, $v$, we get: $$ E = \frac{mc^2}{\sqrt{1-\frac{c^2}{c^2}}} = \frac{mc^2}{\sqrt{1-1}} = \frac{mc^2}{0}\\ E = \infty $$ The reason some particles (such as photons, gluons, gauge bosons, etc...) move at the speed of light (in fact ALWAYS at the speed of light) is becayse they have a rest mass, $m_0$, of $0$ and subsequebtly have a relativistic mass, $m_r$, of indeterminate size (i.e. it can have any amount finite energy). $$ m_r = \frac{m_{0}}{\sqrt{1-\frac{v^2}{c^2}}} = \frac{0}{0} = indeterminate $$

Answer 2

Further to Goodies's answer, there are two easy derivation of why. If you can see how Goodies's answer bars any massive particle from passing through the speed of light, then you need a simple derivation of why Goodies's formula is true.

Method 1: The 4-position of an event in spacetime is $(c\,t,\,x,\,y,\,z)$. To get the four velocity, i.e. something that transforms like a 4-vector by the Lorentz transformation, we need to differentiate it with respect to the proper time $\tau$, where ${\rm d}\tau=\gamma^{-1}\,{\rm d}\,t$, which is the distance between events that all inertial observers agree on, whereas the time between two events depends on the observer. This means that the contravariant four velocity is $\gamma\,(c,\,v_x,\,v_y,\,v_z)$ and the four momentum is $\gamma\,m\,(c,\,v_x,\,v_y,\,v_z)$, where $m$ is the rest mass of a particle. If this is a four vector, then so is $\gamma\,m\,c\,(c,\,v_x,\,v_y,\,v_z)$, and the latter's "length" is $\sqrt{\gamma^2\,m^2\,c^4\,\left(1-\frac{v^2}{c^2}\right)} = m\,c^2$ and it seems reasonable therefore to postulate that the "zeroth" (time) component of the latter four vector is a particle's total energy. Thus we have, for the squared length expression, $E^2 - p^2 c^2 = E^2 - \gamma^2\,v^2 c^2 = m^2\,c^4$, whence the expression in Goodies's answer $E_T = \gamma\,m\,c^2$ and the rest of his answer follows.

Mehtod 2: The following explanation seems a little deprecated in modern physics teaching, but the idea here is think of each little bit of work ${\rm d} W$ done on an accelerating particle by a force appears as an increased inertia of that body equal to ${\rm d} W/c^2$. The increased inertial means that a given increment in speed $\delta\,v$ requires ever more impulse and ever more work, such that if you tally up the total work needed to accelerate a particle to speed $c$, the amount of work diverges to infinity. This is a graphic illustration of why one can't accelerate a massive particle to lightspeed.

Answer 3

If Einstein's 1905 light postulate is false (that is, if the speed of light varies with the speed of the emitter), then the question makes no sense.

"Varies with the speed of the emitter" is equivalent to "varies with the gravitational potential, like the speed of ordinary falling bodies". The Pound-Rebka experiment has confirmed the latter assumption:

http://www.einstein-online.info/spotlights/redshift_white_dwarfs Albert Einstein Institute: "One of the three classical tests for general relativity is the gravitational redshift of light or other forms of electromagnetic radiation. However, in contrast to the other two tests - the gravitational deflection of light and the relativistic perihelion shift -, you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. (...) The gravitational redshift was first measured on earth in 1960-65 by Pound, Rebka, and Snider at Harvard University..."

http://courses.physics.illinois.edu/phys419/sp2013/Lectures/l13.pdf University of Illinois at Urbana-Champaign: "Consider a falling object. ITS SPEED INCREASES AS IT IS FALLING. Hence, if we were to associate a frequency with that object the frequency should increase accordingly as it falls to earth. Because of the equivalence between gravitational and inertial mass, WE SHOULD OBSERVE THE SAME EFFECT FOR LIGHT. So lets shine a light beam from the top of a very tall building. If we can measure the frequency shift as the light beam descends the building, we should be able to discern how gravity affects a falling light beam. This was done by Pound and Rebka in 1960. They shone a light from the top of the Jefferson tower at Harvard and measured the frequency shift. The frequency shift was tiny but in agreement with the theoretical prediction."