Why does it require an infinite amount of energy to reach the speed of light?

[Sagittarius A-Star]

You realize that the tires of a car driving at a paltry 8 kilometers per second will put zero stress on the bridge, right?

With "relativistic speed" I meant significantly faster than 100 km/h, for example 120 km/h.

 

[jbriggs444]

With "relativistic speed" I meant significantly faster than 100 km/h, for example 120 km/h.

By 8 km/s, I am referring to orbital velocity. Relativistic velocities would be some twenty to thirty thousand times faster still. I seriously doubt that you have to worry about the relativistic gamma associated with 120 km/h.

 

[Sagittarius A-Star]

I seriously doubt that you have to worry about the relativistic gamma associated with 120 km/h.

But in front of the bridge, the mass of the cars was only checked at 40 km/h. Maybe, that additional gamma is enough

 

[Sagittarius A-Star]

By 8 km/s, I am referring to orbital velocity.

I wanted to discuss this in the context of SR, according to the thread topic. So I assume a (local) homogeneous gravitational field.

 

[robphy]

In relativity,
I associate tangents with velocities
…. but not like this.

I’m lost how this relates to the OP question about infinite energy to reach the speed of light.

 

[PeterDonis]

I know an example of a scale, that is designed to measure something that is in motion relative to it: In Germany, near Leverkusen, in front of a 50 years old bridge over the river Rhine, https://www.strassen.nrw.de/de/projekte/autobahnausbau-bei-leverkusen/abschnitt-1/lkw-sperranlage.html. All cars have to drive over it with maximum 40 km/h to check, that each car does not have a greater mass than 3500 kg. But I assume that they neglect the gamma-factor, although they measure also the speed.

The reason for the speed limit has nothing to do with relativistic mass. It has to do with the fact that the scale cannot accurately measure the weight of an object moving over it at faster tham 40 km/h because of things like vibration. (Possibly also because of the finite length of the scale, to ensure both that only one car at a time is on it, and that the full length of the car is on it for long enough to make a weight measurement.)

The downward proper acceleration of an object on the surface of a planet like the Earth is independent of tangential velocity, at least for any tangential velocity for which the object remains at the same altitude above the planet for an appreciable length of time. This is a simple consequence of the equivalence principle. So scales don't measure relativistic mass anyway, they measure invariant mass, and therefore your example is off topic anyway.

 

[James Demers]

The difficulty with "relativistic mass" is that it varies with reference frame: as you fly past at some hefty % of c, the fork in your hand may have a relativistic mass of a thousand kilos, in the eyes of a stationary observer. Yet you have no problem eating your pie with it, in your reference frame.
The relativistic mass of the fork has no physical meaning to the stationary observer, unless of course he gets in the way of it. What works just as well in the equations, without creating such apparent paradoxes, is to consider not the relativistic "mass" of the fork, but it's relativistic kinetic energy.

 

[PeroK]

So I think we can agree that the spring would extend the same for a mass $m$ doing $v$ and a stationary mass with a (rest) mass that happened to be $\gamma m$. But we can (and, experience suggests we will) argue about whether that means that you are measuring a relativistic mass of $\gamma m$, or that a spring balance is an inappropriate tool to measure the mass of a body in motion. I tend towards the latter view.

All the issues disappear if you accept the obvious: a scale is fundamentally measuring a force. The measurement of mass is inferred from the force.

 

[jartsa]

To expand on this a bit, apply the equivalence principle and consider a rocket accelerating in flat spacetime. Inside you have a light strong frictionless horizontal rail suspended from a spring, and a small body of mass $m$ moving with (as measured in this frame) constant velocity along the rail. How much force would the spring exert in equilibrium? Since the spring's force is perpendicular to the direction of motion the answer is $\gamma ma$, where $a$ is the "acceleration due to gravity".

So I think we can agree that the spring would extend the same for a mass $m$ doing $v$ and a stationary mass with a (rest) mass that happened to be $\gamma m$. But we can (and, experience suggests we will) argue about whether that means that you are measuring a relativistic mass of $\gamma m$, or that a spring balance is an inappropriate tool to measure the mass of a body in motion. I tend towards the latter view.

Spring balance and mathematical calculations agree about what the force is. But spring balance is wrong tool for measuring the force.

Oh, it was mass that was measured. Well OK then.

The moving object resisted its change of velocity by an extra large force. So how about if we say that moving objects have extra inertia?

 

[vanhees71]

That's correct. Within general relativity inertia and gravity are the same and the sources of the gravitational field are all kinds of energy, momentum, and stress.

 

[PeterDonis]

Moderator's note: Detailed discussion of what a scale measures, based on the specific scenario proposed by @Sagittarius A-Star, has been moved to a separate thread:

https://www.physicsforums.com/threa...ale-in-the-following-rocket-scenario.1005766/

 

[Ibix]

can explain WHY this limit exists

No one knows. It's a postulate of the theory that $c$ is the same in all inertial reference frames. That you can't travel at $c$ follows immediately - because you'd have to have an inertial reference frame where the speed of light is both zero (because you are traveling at the same speed as light) and 3×10⁸m/s (because it is that in every inertial reference frame). But the only justification for the original postulate is post hoc: the predictions we make when we assume it match reality. Why reality is that way, though, no one knows.

 

[MikeWhitfield]

No one knows. It's a postulate of the theory that $c$ is the same in all inertial reference frames. That you can't travel at $c$ follows immediately - because you'd have to have an inertial reference frame where the speed of light is both zero (because you are traveling at the same speed as light) and 3×10⁸m/s (because it is that in every inertial reference frame). But the only justification for the original postulate is post hoc: the predictions we make when we assume it match reality. Why reality is that way, though, no one knows.

That’s what I thought but it seems like several people here felt they understand it well enough to explain it. I had only two years of college physics as part of my engineering curriculum and thus Einstein’s work loses me almost immediately. The math doesn’t look all that different but its level is so far above what I studied that I had trouble just grasping the concepts and flow of logic, let alone understanding any physical significance and underlying phenomena implied by the math. (Haven’t actually tried to study general relativity in a couple decades but IQ tests show that I’m not getting smarter as I age.) I find it fascinating but hold no delusions that I’ll ever truly understand it.

 

[Ibix]

That’s what I thought but it seems like several people here felt they understand it well enough to explain it.

All explanations are based on the maths of special relativity, which ultimately is entirely logical consequences of the two postulates. So if someone says you can't reach light speed because it requires infinite energy you can ask why it requires infinite energy. Keep repeating "why" and you'll eventually get back to the postulates (or something equivalent).

It's fair to say that there will always be a "why" any particular scheme to reach light speed will fail. For example you can't get a rocket to light speed because any amount of fuel provides an energy that corresponds to a speed below lightspeed. Any consistent theory that says you can't do something must provide such explanations for why schemes to do the impossible fail. But you can always ask why...why...why... and ultimately all of those explanations reduce to "it's a consequence of the postulates".

 

[socraticmethead]

And that 5 amount of Newtons increases my speed by 10mph, so I'm now I'm going at 20mph. At this point I stop the rockets and i continue indefinitely at 20mph. So I now apply another 5 Newtons to reach 30mph, and so on until I reach the speed of light

As you go faster, you get heavier (so you need more energy to go faster)
So, 0-10, "five figNewtons"
10-20, "five point, a lot of zeroes, one figNewtons"
20-30... so the faster you go the more figNewtons it requires to go faster again.

e=mc2, the more energy, the more mass. no idea how it works, if the inertia of the mass just increases or what.

E=sqrt(mc^2)^2 + (pc)^2, the parker probe has gone 68,600 m/s, but we're traveling about a tenth of a percent the speed of light, or even more, already. I think I once worked out you need about 6% speed of light for relativistic effects to become easily perceptible (whatever that means), if we imagine we're moving 1% that already, we're 15% there.

First, you need to start studying SR systematically. Posing questions like this that are a mixture of fact, fiction, popular science and your own misconceptions thrown in will get you nowhere.

The first chapter of Morin's book is online here:

https://scholar.harvard.edu/files/david-morin/files/relativity_chap_1.pdf

My personal recommendation is Helliwell's book:

https://www.goodreads.com/book/show/6453378-special-relativity

In partial answer to your question, I would say:

A postulate of special relativity is that the speed of light

 

[PeroK]

As you go faster, you get heavier (so you need more energy to go faster)
So, 0-10, "five figNewtons"
10-20, "five point, a lot of zeroes, one figNewtons"
20-30... so the faster you go the more figNewtons it requires to go faster again.

e=mc2, the more energy, the more mass. no idea how it works, if the inertia of the mass just increases or what.

You're referring to the concept of relativistic mass, which generally is no longer used in teaching SR. E.g. Helliwell has a good summary of its disadvantages. The reason you have no idea how it works is that there is no physical reason for mass to change!

We also have a Insight on it:

https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/

 

[socraticmethead]

You're referring to the concept of relativistic mass, which generally is no longer used in teaching SR. E.g. Helliwell has a good summary of its disadvantages. The reason you have no idea how it works is that there is no physical reason for mass to change!

We also have a Insight on it:

https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/

well I take it back, looks like i do have an idea how it works (it is inertial)

nice insight

 

[Grasshopper]

You're referring to the concept of relativistic mass, which generally is no longer used in teaching SR. E.g. Helliwell has a good summary of its disadvantages. The reason you have no idea how it works is that there is no physical reason for mass to change!

We also have a Insight on it:

https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/

That’s a great Insight, but one thing I’ve wondered is why the concept of relativistic mass was even invented, other than the need to keep equations more familiar. Is there another more useful reason for it being created?

 

[PeroK]

That’s a great Insight, but one thing I’ve wondered is why the concept of relativistic mass was even invented, other than the need to keep equations more familiar. Is there another more useful reason for it being created?

It's not the worst idea, especially when you begin studying SR. The problem is that it can't be extended to GR and it's very clumsy when doing relativistic particle physics generally.

I think Don Lincoln in one of his videos quotes Corinthians in regard of relativistic mass as a childish thing:

When I was a child, I spoke and thought and reasoned as a child. But when I grew up, I put away childish things.

 

[Dale]

It's not the worst idea

It is indeed not the worst, but there are a lot of bad ideas that are quite bad even though they aren’t the worst.

 

[socraticmethead]

I don't need equations, I would just like to pose a question which contradicts the above statement (I know I am wrong btw, I want to see where I am going wrong).

This video explains it well.

Like I said:

for fast things, it's about 1 point a few zeroes 1 figNewtons, or you can convert that to graham-cracker force.

As you get faster - gamma goes higher, momentum = gamma * mass * velocity. this video shows this gamma value - and the effect is on the inertia.

 

[vanhees71]

That’s a great Insight, but one thing I’ve wondered is why the concept of relativistic mass was even invented, other than the need to keep equations more familiar. Is there another more useful reason for it being created?

The concept of relativistic mass was invented in Einstein's very first paper on the subject in 1905. At this time he had not a complete understanding of the dynamics of a point mass. This was clarified pretty soon by Planck. The full understanding of the mathematical structure of special relativity has then be discovered by Minkowski in 1908, and since then all physicists should work in the most simple mathematical formulation, which is to write down covariant laws in Minkowski space. This also makes the physical concepts very clear.

All intrinsic properties of matter are defined in an appropriate rest frame of the matter. For a massive particle it's simply the rest frame of this particle. Then mass is defined in this frame of reference via $E_0=m c^2$. Since energy and momentum together build a four-vector $(p^{\mu})=(E/c,\vec{p})$ you get the relation between energy and momentum in an arbitrary inertial frame by $E=c \sqrt{m^2 c^2+\vec{p}^2}$. In covariant form this "on-shell condition" reads $p_{\mu} p^{\mu}=m^2 c^2$. From this it's also clear that $m$, the invariant mass of the particle, is the same quantity as in Newtonian physics.

In the (for relativity more appropriate) continuum-mechanical description it's the (local) restframe of the fluid/matter cell. That's why all thermodynamical quantities like temperature, chemical potentials, densities are defined as scalars.

Einstein himself also quite early abandoned the concept of "relativistic mass" (you'd even need more than one such "relativistic mass", not only depending on the magnitude of the non-covariant velocity but also on its direction, and that's too complicated to be useful):

https://doi.org/10.1063/1.881171

 

[robphy]

https://doi.org/10.1063/1.881171

From the Physics Today website,
"During 2021, Physics Today is providing complimentary access to its entire 73-year archive to readers who register."From what appears to be (an archive of ) Okun’s website
http://www.itep.ru/science/doctors/okun/publishing_eng/em_3.pdf

From that article,
Okun quotes Einstein:

"It is not good to introduce the concept of the mass M = m/(1 - v^2/c^2 )^{1/2} of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the 'rest mass' m . Instead of introducing M it is better to mention the expression for the momentum and energy of a body in motion." - Einstein (1948)

 

[LBoy]

No one knows. It's a postulate of the theory that $c$ is the same in all inertial reference frames. That you can't travel at $c$ follows immediately - because you'd have to have an inertial reference frame where the speed of light is both zero (because you are traveling at the same speed as light) and 3×10⁸m/s (because it is that in every inertial reference frame). But the only justification for the original postulate is post hoc: the predictions we make when we assume it match reality. Why reality is that way, though, no one knows.

As far as I remember from history of physics this postulate appeared after an attempt to explain the results of experiments of Michelson and Morley, without it the results didn't make sense.

For me it is a direct result of the diferent sign in space-time metrics, for example (-, +, +, +) - there exists "limiting velocity" for every moving body, but I can't say that I understand fully what it means, I can only calculate it without any further physical intuition.

This velocity (often denoted by c) is a part of geometric spacetime structure and it is not a "the velocity of light".

In certain materials light propagates with lesser speeds and other particles (electrons for example) can even exceed the velocity of light there. For reference anyone can find what is Cherenkow radiation. But in these materials there exists the "limiting velocity" as well, so moving electrons there cannot exceed it.

As ar as I remember all Lorentz transformations between reference frames also can be derived from the metric alone (I have it somewhere), so it looks like the half of STW (almost, without mass and energy) comes from the different sign assigned to time-like versor.

 

[LBoy]

To expand on this a bit, apply the equivalence principle and consider a rocket accelerating in flat spacetime. Inside you have a light strong frictionless horizontal rail suspended from a spring, and a small body of mass $m$ moving with (as measured in this frame) constant velocity along the rail. How much force would the spring exert in equilibrium? Since the spring's force is perpendicular to the direction of motion the answer is $\gamma ma$, where $a$ is the "acceleration due to gravity".

So I think we can agree that the spring would extend the same for a mass $m$ doing $v$ and a stationary mass with a (rest) mass that happened to be $\gamma m$. But we can (and, experience suggests we will) argue about whether that means that you are measuring a relativistic mass of $\gamma m$, or that a spring balance is an inappropriate tool to measure the mass of a body in motion. I tend towards the latter view.

I think there may be a cosmic equivalent to this idea.

Take two stars (or two particles), of roughly equal mass and different velocities towards us (as measured by blueshift/redshift for example).

Let them move around some massive object that affects their trajectories. If relativistic mass makes sense (I mean ist is a real mass) then I guess the one with higher speed should be attracted more than the slower one.

I sense there is some mistake I'm making here, but it's the end of a long day and I'm not thinking clearly. I guess what I'm saying is that if velocity is relative and depends on the reference system then mass also depends, so a massive object can't attract two objects with different velocities to us with different force because that would mean that gravity depends on these two masses, massive object and us, measuring velocity of the objects - which doesn't make sense. I'll think it over tomorrow where the problem is here.

 

[jbriggs444]

I sense there is some mistake I'm making here, but it's the end of a long day and I'm not thinking clearly. I guess what I'm saying is that if velocity is relative and depends on the reference system then mass also depends, so a massive object can't attract two objects with different velocities to us with different force because that would mean that gravity depends on these two masses, massive object and us, measuring velocity of the objects - which doesn't make sense. I'll think it over tomorrow where the problem is here.

Yes, you are mistaken here. "Mass" in the sense that the term is normally used is an invariant. It does not depend on the reference system.

If one chooses to define mass differently then it may not be invariant. But then it will not be the same mass the rest of us are talking about. In any case, gravity is not a force in the context of general relativity.

 

[Omega0]

It's not the worst idea, especially when you begin studying SR.

I am not sure about that. I would even say it is a bad idea. For a simple reason: It makes a lot of sense to introduce Lagrangian and Hamiltonian mechanics as soon as possible to "understand" the real laws of nature. It makes no sense to speak about the conservation of mass. It very much makes sense to speak about the conservation of momentum. I believe it is way better to prepare the student for a world which is dominated by energy and time as early as possible. This path allows a student to get into QM easier.
Just my opinion.

 

[vanhees71]

Of course, one should introduce the action principle in Lagrange and Hamiltonian form as soon as possible in the university curriculum. It's impossible to do at the high-school level of course. The most important general subject to be taught early on in physics are symmetry principles, and indeed this should be done in the very first theory lecture on classical mechanics starting with Newton and then introduce also special relativity, of which one approach is to ask, whether Newtonian spacetime is the only possibility to realize the special principle of relativity, i.e., the principle of inertia, and the answer of course is that also Minkowski spacetime is a possibility, and then it's a matter of experiment to answer the question which spacetime concept is better to describe Nature with the obvious answer that it's Minkowski spacetime and thus relativistic physics.

As innocent as it might look, indeed the notion of mass is a problem in this approach, because to really understand it from the symmetry group-theoretical point of view, you need quantum theory. It's also important to keep in mind when teaching classical physics that this is an approximation and QT is the "real thing". So one should be careful not to introduce fundamental concepts that are wrong from the viewpoint of QT, and a relativistic mass is a paradigmatic example for a bad concept, because it doesn't match in any consistent way with the symmetry paradigm as the overarching conceptual framework of all modern physics.

From the point of view of classical Newtonian mechanics mass is pretty enigmatic. It's merely a property describing inertia (in Newton's Lex I) and active and passive gravitational mass (Newton's universal gravitational interaction law). Not to get things wrong from the group-theoretical perspective all you can do, what's anyway natural within Newtonian physics: Just treat it as a scalar to be introduced in finding the Lagrangian of a single particle fulfilling all 10 conservation laws from spacetime geometry, and you end up with $L=m \dot{\vec{x}}^2/2$, where $m$ is a scalar proportionality constant, which turns out to have the meaning of inertial mass when considering also closed systems of 2 and more particles or open systems with particles moving under the influence of "external forces". Newtonian gravity is not a fundamental law but must be empirically introduced leading to the amazing conclusion that all three notions of mass (i.e., inertial, active and passive gravitational masses) are the same (Newtonian equivalence principle).

In this way you have as fundamental concepts of Newtonian mechanics the spacetime symmetries (Galilei symmetry) with the corresponding 10 conservation laws via Noether's theorem. There's no way to derive the conservation of mass, and that's already telling. Nevertheless you need it as another empirical assumption when considering continuum mechanics, where together with an equation of state it's needed to close the equations of motion in addition to the dynamical equations following from the 10 space-time conservation laws.

Further progress in the question of mass is then possible in two ways, depending on the order physics is taught as a whole. In the traditional way the next step is special relativity, and there you can start, as mentioned above, with the question, which spacetime models are possible given the Lex I only. Then you find that not only the Galilei group (and Newtonian spacetime) but also the Poincare group (and Minkowskian spacetime) are possible realizations of this symmetry principle. The very same symmetry analysis for a single particle than naturally leads to the notion of invariant and only invariant mass, i.e., mass stays a scalar quantity, but there's no empirical mass-conservation law anymore, i.e., you can use some example of relativistic reactions of elementary particles like $\mathcal{e}^+ + \mathcal{e}^- \rightarrow \mu^+ + \mu^-$ that clearly doesn't conserve mass but only energy, momentum, and angular momentum. For continuum mechanics instead of mass conservation you need to argue with some other conservation law like electric charge or baryon number together with some equation of state to close the equations.

From the symmetry point of view it's very clear that there's not even a useful idea behind the introduction of a non-covariant quantity called "relativistic mass". It also becomes clear that inertia is not (only) due to mass but due to energy and, as it turns out in the continuum mechanical context, also in some sense momentum and stress. This already hints at drastic changes necessary to also describe gravity. As we all know, it leads to General Relativity, which by construction makes all three kinds of Newtonian masses the same and just to a scalar parameter describing an intrinsic property of matter (as do the other fundamental properties like the different charges of the fundamental interactions described by the Standard Model).

With QT the role of mass becomes clearer, using the symmetry concepts. Here one needs the idea of unitary ray representations, leading to the conclusion that in general a classical symmetry can be realized by analyzing the Lie algebras of the classical spacetime symmetry groups leading then inevitably to the covering group of the corresponding symmetry group, which leads to the conclusion that instead of the rotation group of euclidean 3-space one can consider its covering group SU(2) leading to another intrinsic property of matter, the spin, which can be both integer and half-integer.

Analyzing then the Galilei group further it turns out that the Lie algebra admits a non-trivial central charge, which turns out to be the mass, and interestingly the case of 0 mass corresponding to a unitary representation of the covering group of the classical Galilei group doesn't lead to a useful dynamical quantum theory. So one has to extend the classical Galilei algebra by mass as central charge, leading to a superselection rule excluding superpositions of states belonging to different values of (total) mass, which explains the empirical mass-conservation law within relativistic mechanics.

The same analysis of the proper orthochronous Poincare group for relativistic QT leads to the conclusion that there are no (nontrivial) central charges of it's Lie group and mass is just a Casimir operator of the algebra with no need for a superselection rule nor an additional conservation law for mass in accordance with the empricial observation that in the relativistic realm mass is not conserved. Again there's no hint for why one should introduce an artificial concept of "relativistic masses" at all.

In summary, from a modern point of view thus there's no convincing theoretical or empirical argument for the introduction of a relativistic mass. To my knowledge there's not even a useful heuristic argument to introduce it!

 

[Dale]

The invariant mass of an isolated system is conserved. It is not additive but it is conserved.

 

[vanhees71]

It depends how you define "isolated system". E.g., some piece of matter gets a larger invariant mass when it gets warmer or a charged capacitor has a larger invariant mass than an uncharged one etc.