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

[Ibix]

Why not? A photon has constant velocity why can't we assign an inertial frame to it?

The second postulate says that the speed of light is the same in all inertial frames of reference. A rest frame for light is, therefore, a contradiction in relativity - light cannot be doing $c$ and be at rest at the same time.

 

[vanhees71]

You can NOT learn physics without equations. There are tons of constructive answers in this thread!

 

[jartsa]

Thank you all for your answers. So one thing I'd just like to clarify (as everyone keeps saying) is that - your mass does not increase as you accelerate? Or is this a question of it depends on who is observing? As in, to me moving at the same speed (or even accelerating) my mass is the same as it always has been; but to another observer my mass is increasing proportionately to the rate at which I am accelerating, from their reference frame?

Does the velocity of the object whose relativistic mass we are talking about change?

If the velocity changes, then the relativistic mass changes.
If the velocity does not change, then the relativistic mass does not change.

(I mean the old relativistic mass that does not exist in modern physics )

 

[vanhees71]

Have you considered that our observable 3 dimensions are actually 6 and the vector you talk about may not be what one thinks?

ps: mathematically don’t we know there are 10-11 dimensions? I’m just hoping one of you gurus put this in mathematical terms

Ok, now finally I must give the mathematical description, no matter whether you want it or not.

You start with four-vector components $(x^{\mu})=(ct,\vec{x})$ for time and space (the spacetime four-vector). The motion of a particle is described as a world line in this four-dimensional vector space. For massive particles this world line must be time-like, and thus you can choose proper time as the world-line parameter. This is the time measured by an ideal clock co-moving with the particle. It is defined by
$$\mathrm{d} \tau=\sqrt{\mathrm{d} t^2-\mathrm{d} \vec{x}^2/c^2}.$$
It is thus related to the coordinate time wrt. the inertial frame used to do the calulation by
$$\frac{\mathrm{d} \tau}{\mathrm{d} t}=\sqrt{1-(\mathrm{d}_t \vec{x})^2/c^2}=\sqrt{1-\vec{v}^2/c^2}=1/\gamma.$$
In order to have a covariant description one defines a four-vector
$$p^{\mu}=m \mathrm{d}_{\tau} x^{\mu}.$$
Since $x^{\mu}$ is a four-vector and $\mathrm{\tau}$ is a scalar, $m$ necessarily is a scalar too in order to have $p^{\mu}$ as a four-vector.

Expressing $p^{\mu}$ in terms of coordinate-time derivatives gives
$$(p^{\mu})=m \gamma \begin{pmatrix}c \\ \vec{v} \end{pmatrix}.$$
In an inertial frame, where $|\vec{v}| \ll c$ you have $\gamma=1+\vec{v}^2/(2 c^2)+\mathcal{O}(v^4/c^4)$ and thus
$$(p^{\mu}) \simeq \begin{pmatrix} m c +m v^2/(2 c)+\cdots \\ m \vec{v} + \cdots \end{pmatrix}.$$
This shows that
$$p^0=m c + E_{\text{kin}}/c,$$
and $\vec{p}$ takes the same form as in Newtonian physics with $m$ the usual mass known from Newtonian physics.

The relativistic connection between energy and momentum, where energy is defined such that it includes the socalled rest energy $E_0=m c^2$, thus is
$$E=m \gamma c^2=\frac{m c^2}{\sqrt{1-v^2/c^2}}, \quad \vec{p}=m \gamma \vec{v}=\frac{m \vec{v}}{1-v^2/c^2}.$$
This shows that necessarily $|v|<c$ and to reach the limit $v \rightarrow c$ you need an infinite amount of energy.

There are no 6 dimensions nor 10-11 dimensions in standard relativistic theory. Also Minkowski space is a real 4D vector space. One should avoid textbooks using the ancient $\mathrm{i} c t$ convention, because it is quite confusing and also cannot be extended to noninertial reference frames in SR, let alone to GR, where you work with arbitrary spacetime coordinates anyway.

You find some introduction to special relativity in my (still unfinished) manuscript

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

There you also find, how to covariantly formulate classical electrodynamics.

 

[mfb]

I deleted several non-constructive posts and replies to them.

 

[Grasshopper]

Thank you all for your answers. So one thing I'd just like to clarify (as everyone keeps saying) is that - your mass does not increase as you accelerate? Or is this a question of it depends on who is observing? As in, to me moving at the same speed (or even accelerating) my mass is the same as it always has been; but to another observer my mass is increasing proportionately to the rate at which I am accelerating, from their reference frame?

The way I look at it: Physicists at the time invented a new definition of mass that was velocity dependent so that they could hold on to the p = mv form for momentum. All it really is is the relativistic momentum equation divided by v.

 

[Mister T]

Are we all not moving at the speed of light relative to a photon or a neutrino?

It was discovered several years ago that neutrinos have a small nonzero mass, thus they do not travel at speed $c$.

 

[Sagittarius A-Star]

to another observer my mass is increasing proportionately to the rate at which I am accelerating, from their reference frame?

No. Also, your description is by far not as precise as it would be, if it had been formulated as an equation. You mean the "relativistic mass":
$m_R = m_0\frac {1}{\sqrt{1-v^2/c^2}}$
As others have already proposed, it is not good to use the "relativistic mass", because it is redundant to energy and therefore needless. Better call it $\frac{E}{c^2}$.

Then you see also directly the answer to your question in the O.P. As $v$ approaches $c$, the energy increases without bound.

In classical mechanics, mass is the quantity of matter. In SR, a quantity of matter does not exist. It is the energy of an object, that is inertial. It includes all kinds of energy, for example thermal energy. If you stand on a bathroom-scale, then you measure your energy content.

 

[PeroK]

If you stand on a bathroom-scale, then you measure your energy content.

That's measuring a force, surely.

 

[sysprog]

That's measuring a force, surely.

The tension in the spring?

 

[Sagittarius A-Star]

That's measuring a force, surely.

On the display of the scale stands normally the unit $kg$, but is could be also $Ws$.
$F = \frac{E}{c^2} \cdot 9.81 m/s^2$

 

[Mister T]

That's measuring a force, surely.

No, it's measuring mass. When precision is required, those scales are calibrated for the location in which their use is intended.

 

[jbriggs444]

We should probably shy away from turning this thread into a further argument about what scales measure.

One thing is fairly clear. If you are standing on the scale then we need not concern ourselves with whether it is measuring relativistic or invariant mass (or local g times either). The two sorts of mass coincide for objects with zero total momentum.

 

[Sagittarius A-Star]

One thing is fairly clear. If you are standing on the scale then we need not concern ourselves with whether it is measuring relativistic or invariant mass (or local g times either). The two sorts of mass coincide for objects with zero total momentum.

Yes. And if you are moving with constant horizontal velocity $v$ on the scale while the measurement, then is measures transversal mass energy $E=\gamma E_0$.

 

[DrGreg]

Yes. And if you are moving with constant horizontal velocity $v$ on the scale while the measurement, then is measures transversal mass energy $E=\gamma E_0$.

No, a scale isn't designed to accurately measure anything that is in motion relative to it (especially if the speed is relativistic).

 

[Ibix]

No, a scale isn't designed to accurately measure anything that is in motion relative to it (especially if the speed is relativistic).

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.

 

[Sagittarius A-Star]

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 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.

 

[DaveC426913]

...I assume, that they neglect the gamma-factor...

 

[Sagittarius A-Star]

I fear that the bridge will break, if the cars drive with relativistic speed over it (because of the high gamma-factor).

 

[jbriggs444]

I fear that the bridge will break, if the cars drive with relativistic speed over it (because of the high gamma-factor).

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

 

[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