Exactly Why C is impossible for Massive Objects?

[AndromedaRXJ]

Everytime I look this up, the answer always is "because mass increases as you approach C, therefore infinite energy is required.", but from what I understand, Relativistic Mass is a misconception and there's no actual increase in mass.

So if infinite energy is required, then why? And why else is it impossible to reach C with mass?

 

[pervect]

You don't need dynamics (i.e masses, forces, etc) to determine that it's impossible to reach 'c'. All you need is kinematics. In case this term is unfamiliar, I'll copy the definition from wiki.

Kinematics is the branch of classical mechanics that describes the motion of points, bodies (objects) and systems of bodies (groups of objects) without consideration of the causes of motion.

Suppose you have an series of numbered objects 1,2,3,4,5,6,7...etc, where the number of the last object is finite, but as large as you like.

Now suppose the speed of each object is 1 meter/second in the frame of the object before it.

I.e. the speed of object 2 is 1 m/s in the frame of object 1, the speed of object 3 is 1 m/2 in the frame of object 2, the speed of object 100 is 1 m/s in the frame of object 99.

When you use the relativistic velocity addition laws, you will find that the speed of object n relative to object 1 does not grow without limit, but instead approaches 'c' , as n increases without bound.

The relativistic velocity laws say that v1+v2 = (v1+v2)/(1+v2*v2/c^2)

I don't currently have a closed form expression for the series summation of this, but it's easy to show that if v1 and v2 are less than c, the sum v1+v2 is less than c.

It might be interesting and instructive to find (or attempt to find, at least) a closed form solution for the series.

You can look at dynamics and at energy if you like in that case you just use the relativistic expressions for energy, E = gamma*m, m being the invariant mass.

 

[Bill_K]

The relativistic velocity laws say that v1+v2 = (v1+v2)/(1+v2*v2/c^2)
I don't currently have a closed form expression for the series summation of this, but it's easy to show that if v1 and v2 are less than c, the sum v1+v2 is less than c.

It's the formula for tanh(a + b), right? tanh(a + b) = (tanh a + tanh b)/(1 + tanh a tanh b).

So let v be the relative velocity between object n and object n+1, and let a = tanh^-1(v). Also let b_n = tanh^-1(v_n).

Then the recursion relation becomes b_n+1 = a + b_n, so b_n = na, and v_n = tanh(n tanh^-1(v)).

 

[Naty1]

but from what I understand, Relativistic Mass is a misconception and there's no actual increase in mass.

Well, some would say that, maybe, but it's a subject of controversy in these forums...

For a discussion of the pros and cons see here:

On the concept of Mass
https://www.physicsforums.com/showthread.php?t=696144&highlight=relativistic+massWikipedia points out Okun, and Taylor and Wheeler don't like the concept:

http://en.wikipedia.org/wiki/Relativistic_mass#Controversy

So if infinite energy is required, then why?

You can see why people say this from the first formula in the 'Controversy' section above...which happens to be for momentum, but the same idea holds... when you have the Lorentz
factor 1/ sq rt[1 -v²/c²] as v approaches c, the expression diverges.

And why else is it impossible to reach C with mass?

The easiest way to think about this: If the speed of light is 'c' in all inertial frames, how could you ever catch up?? Even at .9999c a mass observer would see light whizzing by locally at good old 'c'.
Of course the fact that every inertial frame observer sees light at 'c' has no explanation other than it is observed. There is no set of first principles that demands that and excludes all other possibilities. It's the way stuff works in this universe.

Edit: I see Bill_k posted while I was composing...the hyperbolic functions he mention derive from the Lorentz relationship I posted...two sides of the same coin...

You can fuss with the math from here if desired:

http://en.wikipedia.org/wiki/Lorentz_factor

See the first formula and then its relationship to hyperbolic functions under RAPIDITY.

 

[DrGreg]

In fact, you don't need to know any mathematical formulas at all, you just need to know Einstein's 2nd postulate. In pervect's example in post #2, object 1 measures the speed of light relative to itself and gets the answer c. Object 2 measures the speed of light relative to itself and, by the 2nd postulate, must also get the same answer c. And so must object 3. And so must object 4. And so on, for all the objects. None of the objects are "getting any closer" to the speed of light, according to their own measurements.

 

[DrGreg]

v_n = tanh(n tanh^-1(v)).

Just to be pedantic, and for the benefit of readers not familiar with the convention of using units where $c = 1$, that would be$$v_n = c \, \tanh \left( n \, \tanh^{-1}\frac{v}{c}\right)$$

$v_n \rightarrow c$ as $n \rightarrow \infty$.

 

[nitsuj]

How come it's not the geometry first and all else follows?

The "rules" of spacetime are the postulates, "drilling down" to the concept of c, specifically how length and time are defined limits the speed; to what ever magnitude. Further with that, it's causality.

So speed is limited, "invariantly", because one thing leads to another; and the whole universe has to "see" this physically "play out" the same. We measure these gaps across spacetime between "happenings" using time and length. now reading the underlined part above helps

It seems strange because we move so slow compared to things around us, but speed is really something physically fundamental to the causal system as a whole (Universe), as fundamental as the "every action there is an equal but opposite reaction" we are more familiar with. When we examine this very fundamental perspective, it's separated it into two measures. That makes sense in a Newtonian world, since we could all agree on the measured results. In SR the measurement across space time is never just time or length but a perfect balance of the two if you are inertial. consider measuring time/length where there is gravity/tidal gravity. With gravity potential you are not inertial and comparative measures of length / time from different potentials would not be exact.

I like that question "And why else is it impossible to reach C with mass? ".

lol because gravity can only go c, you can break the sound barrier but not the causality barrier. In fact if you ever leave your personal gravity bubble you'd be in trouble, so don't even think of it!

Wow what a neat perspective, just by "being/existing " matter is causally connected at c via gravity, hmmm as of my birth my gravity field would be nearly 66 light years across today...now that's quite a presence! :rofl: I wonder what that would be cubed for volume. And what physical property is attributed to that value. Oh right it's just fictitious lol gravity, pfft Who's the mathematician that ruined GR? if gravity is fictious so is speed.

 

[pervect]

Well, some would say that, maybe, but it's a subject of controversy in these forums...

Last I looked most if not all of the Science Advisors and Mentors recommended against relativistic mass. Not much of a "controversy" in my opinion.

It's not IMPOSSIBLE to use relativistic mass correctly, but rare. "Energy" is a direct synonym for it, and when "relativistic mass" is called "energy" it seems to be misused much less frequently.

I'd preach a little more but I feel like it won't actually help stem the tide of elementary misunderstandings that seem to be highly correlated with use of the term, so I'll leave it at that.

 

[pervect]

It's the formula for tanh(a + b), right? tanh(a + b) = (tanh a + tanh b)/(1 + tanh a tanh b).

So let v be the relative velocity between object n and object n+1, and let a = tanh^-1(v). Also let b_n = tanh^-1(v_n).

Then the recursion relation becomes b_n+1 = a + b_n, so b_n = na, and v_n = tanh(n tanh^-1(v)).

Just to be pedantic, and for the benefit of readers not familiar with the convention of using units where $c = 1$, that would be$$v_n = c \, \tanh \left( n \, \tanh^{-1}\frac{v}{c}\right)$$

$v_n \rightarrow c$ as $n \rightarrow \infty$.

Ah yes, if one remembers that rapidities add, the answer becomes clear.

 

[Vanadium 50]

Last I looked most if not all of the Science Advisors and Mentors recommended against relativistic mass. Not much of a "controversy" in my opinion.

If you'll do a search, you'll also see that the "controversy" has one long-term member who supports it. And a bunch of crackpots, of course. Injecting this does absolutely nothing to help the OP understand.

AndromedaRXJ, Science Advisors have a long track record of correct and helpful posts. You probably want to keep that in mind.

The easiest way I think there is to show this is to calculate the velocity of an object with mass m and energy E:

$$v = c \frac{\sqrt{E^2-m^2c^4}}{E}$$

Note that the numerator is always less than E, so the velocity is always less than c.

 

[Raze]

If you'll do a search, you'll also see that the "controversy" has one long-term member who supports it. And a bunch of crackpots, of course. Injecting this does absolutely nothing to help the OP understand.

AndromedaRXJ, Science Advisors have a long track record of correct and helpful posts. You probably want to keep that in mind.

The easiest way I think there is to show this is to calculate the velocity of an object with mass m and energy E:

$$v = c \frac{\sqrt{E^2-m^2c^4}}{E}$$

Note that the numerator is always less than E, so the velocity is always less than c.

I know I really shouldn't ask this, but what if m was imaginary? Then wouldn't you always have a velocity greater than c? (sorry for the off topic but I read something on wikipedia about tachyons having this property some time ago)

 

[PAllen]

I know I really shouldn't ask this, but what if m was imaginary? Then wouldn't you always have a velocity greater than c? (sorry for the off topic but I read something on wikipedia about tachyons having this property some time ago)

That's the definition of a tachyon: imaginary mass, postive energy, speed > c. They were hypothesized, searched for and not found. A majority of physicists (but by no means a near universal consensus) believes they are unlikely to exist. They produce causality problems given certain reasonable assumptions.

 

[anuraj.b]

velocity and quantum particles

In quantum physics the particles move with velocity c(=velocity of light) , each particle energy is quantised. E=nhc/λ. this particles called quantum particles.
Where n is an integer.
consider massive bodies there is infinite number of quantum particles inside the nucleus and they are bonded with each other. so massive body doesn't move with the velocity c
the reason is quantum particle are bonded due to nuclear forces,columb force etc. if these particles are separately each particle move with the velocity c.

 

[nikkkom]

So if infinite energy is required, then why? And why else is it impossible to reach C with mass?

Formulas of special relativity's Lorenz transform show that however you increase the velocity, v+dv will never reach c.

Mathematical models don't answer the question "why?".

If you are asking "why do we use Lorenz transform?", the answer is "because it matches observations".

 

[harrylin]

Everytime I look this up, the answer always is "because mass increases as you approach C, therefore infinite energy is required.", but from what I understand, Relativistic Mass is a misconception and there's no actual increase in mass.

So if infinite energy is required, then why? And why else is it impossible to reach C with mass?

No matter how you name it, the increased kinetic energy brings with it an increase of resistance against acceleration - even to acceleration perpendicular to its motion. And for acceleration in the direction of motion additional work must be done for the increase in kinetic energy, which is disproportional to the increase in speed as shown by the equation in post #10 by Vanadium.

 

[Vanadium 50]

I know I really shouldn't ask this, but what if m was imaginary

Let's not hijack this thread. The question was about massive particles, not hypothetical particles with imaginary masses.

In quantum physics the particles move with velocity c(=velocity of light)

That's not true.

Let's try and answer the question the OP asked, OK?

 

[mfb]

consider massive bodies there is infinite number of quantum particles inside the nucleus

Can you provide a source for that claim?

and they are bonded with each other. so massive body doesn't move with the velocity c

I doubt that you can draw that conclusion like that.
The proton mass, for example, is a fixed value as well, and if you fix its wavelength (like you did for light), you get a similar expression for the energy.

if these particles are separately each particle move with the velocity c.

That is wrong, electrons and quarks have rest masses.

 

[AndromedaRXJ]

Thank you everyone, for responding to my question.

So let me see if I got this down. Mass particles can't reach c simply because c is constant in all inertial frames. And, through experimental observation, we find that length contracts, time dilates and simultaneity is relative, which preserve this fact about c. And as a particle approaches c, these relativistic effects are extreme just enough to prevent a particle from reaching c.

Why the universe is this way is unknown, but we know that it simply just is(through experiments). Is that right?

No matter how you name it, the increased kinetic energy brings with it an increase of resistance against acceleration

Okay, so help me to understand this resistance to accelerate.

Is this more to do with what's going on with the space-time around the mass approaching c, rather than the internal structure of the mass? Like length contracts just enough to where each change in speed requires proportionally more energy?

That's what I got out of the discussion that Naty1 posted a link to.

even to acceleration perpendicular to its motion.

Wait, really? So our change in velocity becomes harder?

So if we travel in a circle at constant speed of 0.99c, the rate of energy expense is needed to steadily go up to stay in the circle? Or am I misunderstanding?

What about decelerating?

 

[mfb]

Why the universe is this way is unknown, but we know that it simply just is(through experiments). Is that right?

Right

Is this more to do with what's going on with the space-time around the mass approaching c, rather than the internal structure of the mass? Like length contracts just enough to where each change in speed requires proportionally more energy?

Length is not a property of space. It is a measurement you do. In special relativity, you do not change spacetime at all - you just use it to make your measurements in.

Wait, really? So our change in velocity becomes harder?

So if we travel in a circle at constant speed of 0.99c, the rate of energy expense is needed to steadily go up to stay in the circle? Or am I misunderstanding?

The required force grows. You do not need energy to keep a particle in a circle.*

What about decelerating?

Needs more force (and gives more energy), too.*unless you consider synchrotron radiation, but that is beyond the scope of this thread.

 

[nitsuj]

Why the universe is this way is unknown, but we know that it simply just is(through experiments). Is that right?

Right

I'd beg to differ, it's kinda clear why it would be this way, especially if some time is spent with Pythagoras theorem, understanding 4D & causality.

 

[AndromedaRXJ]

The required force grows.

So going at a constant speed of 0.99c in a circle, the force needed to maintain the circular path constantly GROWS?

Like, for instance, a tether keeping us in the circle would have to constantly grow in tensile strength, AS IF the mass of the object is increasing, classically speaking?

 

[PAllen]

So going at a constant speed of 0.99c in a circle, the force needed to maintain the circular path constantly GROWS?

Like, for instance, a tether keeping us in the circle would have to constantly grow in tensile strength, AS IF the mass of the object is increasing, classically speaking?

Yes. A better way to look at it, though, is momentum increases with minimal increase in speed. Force is really dp/dt. So momentum keeps increasing as force is applied, but the relation between momentum and speed in SR is very different from Newtonian physics.

 

[pervect]

$v_n = c \, \tanh \left( n \, \tanh^{-1}\frac{v}{c}\right)$

I was thinking it might be helpful to simplify this a bit more. Setting v=1 m/s, as in my original example, we can write

$v_n \approx c \, (\tanh 3.335641 \,10^{-9} (n-1))$

(to avoid any concerns about the approximation, we could also write)
$c \, \tanh (3.33564\,10^{-9} (n-1)) < v_n < c \, (\tanh 3.335641 \,10^{-9} (n-1))$

where c = 299792458 m/s

and I"ve chosen my indexing such that the velocity of spaceship 1 is zero and the velocity of spaceship 2 is 1, hence the n-1.

This makes it really clear, I hope, that in the limit as n->$\infty$, v_n goes to c * tanh($\infty$) = c and that v_n is always lower than c for any finite n.

We didn't need any consideration of forces , masses, or any other dynamical elements to compute this - all we needed to know was the relativistic velocity addition formula.

 

[Samshorn]

You don't need dynamics (i.e masses, forces, etc) to determine that it's impossible to reach 'c'. All you need is kinematics... suppose the speed of each object is 1 meter/second in the frame of the object before it...

That's self-contradictory. The very definition of a "frame" (by which you must mean a standard inertial frame, since otherwise the "speeds" defined in terms of it need not satisfy that composition formula) is based on the dynamics and inertia of physical entities. The relativistic velocity composition formula is a consequence of how inertial coordinate systems are related to each other, which in turn is a consequence of the fact that all forms of energy - including kinetic energy - have inertia (something which Newton didn't realize).

We didn't need any consideration of forces , masses, or any other dynamical elements to compute this - all we needed to know was the relativistic velocity addition formula...

But the physical meaning of the relativistic velocity composition formula (including the identification of the symbol "v" with a physical "speed") is based entirely on the dynamical inertia of physical objects and processes, which is the basis for the definitions of frames and speeds. The answer to the OP's question is that all forms of energy, including kinetic energy, have inertia. It isn't possible to give a meaningful account of special relativity without reference to inertia.

 

[harrylin]

[..] Okay, so help me to understand this resistance to accelerate.

Is this more to do with what's going on with the space-time around the mass approaching c, rather than the internal structure of the mass? Like length contracts just enough to where each change in speed requires proportionally more energy?

That's what I got out of the discussion that Naty1 posted a link to.

Space-time is hardly affected - according to SR, not at all (it's a GR correction). Instead, the energy is carried along with the object, and at impact this energy is given off as heat (this was shown in a famous demonstration experiment by Bartocci).

Wait, really? So our change in velocity becomes harder?

Yes, in a particle accelerator* more force is needed to laterally accelerate a particle when it moves at high speed: its momentum p = γ m₀ v. Thus it is sometimes said that the particle becomes more "massive" at higher speed, as it has increased inertia: it is harder to deflect.

*https://en.wikipedia.org/wiki/Particle_accelerator

So if we travel in a circle at constant speed of 0.99c, the rate of energy expense is needed to steadily go up to stay in the circle? Or am I misunderstanding?
[..]

Neglecting radiation, no energy is needed ("no work is done") to keep a particle in circular motion; it is just harder to pull it inwards than one would expect from Newton's laws.

 

[harrylin]

[..] the physical meaning of the relativistic velocity composition formula (including the identification of the symbol "v" with a physical "speed") is based entirely on the dynamical inertia of physical objects and processes, which is the basis for the definitions of frames and speeds. The answer to the OP's question is that all forms of energy, including kinetic energy, have inertia. It isn't possible to give a meaningful account of special relativity without reference to inertia.

I largely agree with you, but kinematics vs dynamics as cause is clearly a matter of opinion. Although the very definitions of those words give support for our opinion, it is the topic of a never ending debate. See for example http://philsci-archive.pitt.edu/3895/1/kin-vs-dyn.pdf.

For this thread it would be a bad idea to mirror that debate; however it may be useful for the OP to present some of the explanations that have been given in the course of that debate.

PS: Likely the OP agrees with Einstein (and Brown and Janssen and myself) that "When we say that we have succeeded in understanding a group of natural processes, we invariably mean that a constructive theory has been found which covers the processes in question". - Einstein 1919, as cited in the above-mentioned article.

 

[pervect]

That's self-contradictory. The very definition of a "frame" (by which you must mean a standard inertial frame, since otherwise the "speeds" defined in terms of it need not satisfy that composition formula) is based on the dynamics and inertia of physical entities.

Whose definition would that be?

I don't see any self contradiction in saying that one can talk about positions and their rate of change of position with respect to time (velocities) without reference to the masses or forces acting on the bodies (dynamics). That's basically the textbook definition of kinematics, which I quoted earlier. I looked up a few other sources to check, they say much the same thing.

I'm not convinced that what I said contradicts any external definition either, but if you have a reference that you can quote that offers a contradictory definition, I'm willing to examine the matter further. At the moment, however, I must say that I just don't see your point at all, not even a teensy tiny little bit :-(.

Early relativity courses talk about the kinematics first (the description of position and velocities, and how they transform). Later on in the course dynamics will often be introduced. But one does not need the more advanced concepts introduced in the relativistic dynamics to appreciate that bodies can't exceed the speed of light.

 

[mfb]

So going at a constant speed of 0.99c in a circle, the force needed to maintain the circular path constantly GROWS?

Like, for instance, a tether keeping us in the circle would have to constantly grow in tensile strength, AS IF the mass of the object is increasing, classically speaking?

Wait, wait...
The required force grows with increasing velocity, not with time.
To keep a particle in a circle at .99c needs more force than keeping a particle in a circle at .9c (a factor of 3 between the required forces). That is what I meant with "growing".
To keep a particle in a circle at a constant speed over some time needs a constant force*.

*well, the direction of the force has to change with time, of course, but the magnitude stays the same.

 

[AndromedaRXJ]

So then inertia only increases with a change in speed but not a change in velocity in Special Relativity, right?

I'm trying to understand where the need for infinite energy(or force) comes from to reach c.

 

[nitsuj]

I'm trying to understand where the need for infinite energy(or force) comes from to reach c.

not being able to reach c comes first, what follows is the measurements/calculations. For example, does it even makes sense to say "it would require infinite energy to reach c"? The question is; why mass cannot "go" exactly c?

Makes me wonder if when calculating the "energy" of a wave with classical thinking, was the result infinite energy?

Is algebra math to Maxwell math a chasm?

 

[AndromedaRXJ]

not being able to reach c comes first, what follows is the measurements/calculations. For example, does it even makes sense to say "it would require infinite energy to reach c"? The question is; why mass cannot "go" exactly c?

That did cross my mind, but I guess I thought it was acceptable to ask it that way.

Rephrasing the question; why does any finite amount of energy(or force) applied to the particle always result in a speed less than c? And is it energy or force? And does this only apply to a change in speed but not necessarily a change in velocity?

 

[mfb]

So then inertia only increases with a change in speed but not a change in velocity in Special Relativity, right?

A change in speed is always a change in velocity.

I'm trying to understand where the need for infinite energy(or force) comes from to reach c.

$$E=\gamma m c^2 = \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}$$ As v approaches c, the energy grows without bounds. In other words, every finite energy corresponds to a velocity below c.

You can derive similar equations like $F=m \gamma^3 a$ or, equivalently, $a=\frac{F}{m \gamma^3}$. As the velocity increases, γ increases so quickly that you never reach c, independent of the force and acceleration track you have.

 

[nitsuj]

That did cross my mind, but I guess I thought it was acceptable to ask it that way.

Rephrasing the question; why does any finite amount of energy(or force) applied to the particle always result in a speed less than c? And is it energy or force? And does this only apply to a change in speed but not necessarily a change in velocity?

It totally is "acceptable" to ask it the way you did. But it is important to "see" the perspective of the inquiry.

I wish I could answer the question from the perspective you are asking, I only "understand" SR geometrically. I don't know the energy / mass equivalence well enough yet to explain c with it the same way I can with length/time. Saying mass "stays still-length" and energy "moves-time" doesn't clarify anything lol,

 

[Samshorn]

The very definition of a "frame" (by which you must mean a standard inertial frame, since otherwise the "speeds" defined in terms of it need not satisfy that composition formula) is based on the dynamics and inertia of physical entities.

Whose definition would that be?

Yours, I would think. You're not claiming you can define an inertial frame without reference to inertia (dynamics), are you?

I don't see any self contradiction in saying that one can talk about positions and their rate of change of position with respect to time (velocities) without reference to the masses or forces acting on the bodies (dynamics).

Again, the velocities you're talking about are defined only in terms of an inertial frame, which is defined based on the inertial behavior (dynamics) of physical objects and phenomena. Without that connection, there is no physical meaning to equations such as the velocity composition formula. (Obviously that formula doesn't apply to any arbitrary kinematically defined system of coordinates.)

Early relativity courses talk about the kinematics first (the description of position and velocities, and how they transform). Later on in the course dynamics will often be introduced.

Those courses are just witlessly mirroring a well-known malapropism in Einstein's 1905 presentation, where he begins with a section labeled "Kinematical Part", and yet the very first sentence is "Let us take a system of coordinates in which the equations of Newtonian mechanics hold good", and the entire discussion that follows consists of statements that are true only on the basis of those dynamically defined inertial coordinates. Indeed that operational dynamical basis is the only thing that gives physical meaning to those statements. Thus the entire discussion is based squarely on mechanics (dynamics). What he meant by "kinematical" was really just that in this section he establishes the systems of measure for space and time, not that these were to be defined without reference to physical phenomena (any more than they are in Newtonian physics), which would be absurd.

Kinematically we can say the Sun goes around the Earth; it is only in terms of inertial coordinates that we can say unequivocally the Earth goes around the Sun (more or less). Kinematically the two twins are on a symmetrical footing - they separate and come back together; it is only by distinguishing between inertial and non-inertial paths that we can say one twin ages more than the other. The speed of light can obviously take arbitrary values in terms of arbitrary kinematic coordinate systems; it is only in terms of an inertial coordinate system that we can say the speed of light is c.

You expressed your "kinematical" statements in terms of speeds defined in terms of inertial frames, so I suggest you think about what you really mean by "speed" and "inertial frame", and if it's possible to give them physical meaning without invoking dynamics and inertia. (It isn't.)

 

[AndromedaRXJ]

A change in speed is always a change in velocity.

I know, but a change in velocity doesn't have to be a change in speed.

That's why I brought up the particle going in a circle at a constant speed of 0.99c. Is it's inertia growing due to it's circular path?

$$E=\gamma m c^2 = \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}$$ As v approaches c, the energy grows without bounds. In other words, every finite energy corresponds to a velocity below c.

You can derive similar equations like $F=m \gamma^3 a$ or, equivalently, $a=\frac{F}{m \gamma^3}$. As the velocity increases, γ increases so quickly that you never reach c, independent of the force and acceleration track you have.

So would γ increase to a change in direction?