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

[Mister T]

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. This does not require an infinite amount of energy.

Equal amounts of thrust do not result in equal increases in speed.

 

[PeterDonis]

I was trying to show that the statements I read about reaching the speed of light don't make sense if speed is relative.

The speed of light is not relative. It is the same in all reference frames. Only speeds slower than light are relative, in the sense that they change when you change reference frames. The speed of light does not.

 

[mucker]

This already excludes any understanding of physics. If you don't need (or want) equations (math) you don't need to waste your time in trying to understand any modern physics, including special (let alone general) relativity

I wasn't trying to imply I am beyond knowing the math. I think you have interpreted this as some sort of arrogance on my part. I didn't give any context to that statement so I can see why you may take it that way. The reason I started out with that statement is because I won't understand the math. So where I was coming from is that I would prefer it if we kept the math out of the answers as I simply won't "get it".

Let me give an analogy of what I mean. When you are answering questions you can give a "high level" overview without going into the details, this is what I was asking for (if possible). As an example of what I mean - let's say I am programmer and you are not IT literate at all and ask me what my program does. I can explain to you what it does and go into detail without showing you the code (aka math formulas). Even if the program is complicated (like SR) I can explain how it works along the lines of "x does this etc, just trust me). You can still understand (maybe that is a strong word in this context, maybe "make sense of" is better way to put it) the program (SR or GR) and "get it", but to prove it I'd need to show you the code (maths) and YOU would need to understand the programming language (understand the equations) to verify it. What I am essentially saying is that right now I don't need the verification part (as I can't grasp it yet), I will just trust what you say as true, so adding the math only complicates it more without any benefit at this point. I understand that to fully grasp SR and GR I need to understand the math - but right now none of it will make any sense, but talking in plain English will...

I just wanted to clear that up in case I came across as some arrogant fool that thinks they don’t need to lower themself to the math.

 

[Omega0]

Oh it doesn't, no need for unlimited energy to reach the speed of light. Shown by photons pretty often. About the other case, if you got the math (as you say) I personally would recommend to observe the nature which is the basic for physics. If since more than 100 years we didn't made a massive particle faster than light and the SRT works so far very well I personally would search other realms in the physics world to critizise.

 

[anuttarasammyak]

Coming back to OP post #1 by mucker

A spaceship is in inertial motion with speed 0 wrt to the Earth which it is at rest in IFR#0.
After consuming energy E, the spaceship is in inertial motion with speed v wrt to the Earth. Say it is at rest in IFR#1.
After consuming energy E, the spaceship is in inertial motion with speed v in IFR#1. Say he is at rest in IFR#2.
After consuming energy E, the spaceship is in inertial motion with speed v in IFR#2. Say he is at rest in IFR#3.
...
After consuming energy E, the spaceship is in inertial motion with speed v in IFR#n. Say he is at rest in IFR# n+1.
...

In this consequence with knowledge of Galilean transformation Earth people anticipate to observe that spaceship model n have speed of
$$v+v+v+v+v+...$$
to easily surpass c in finite n with energy consumption of nE. But the reality is we need infinite n to reach c.

Many people including mucker claim it is due to nature of M but it actually due to spacetime structure denying simple velocity addition rule. It is not right to keep Galilean transformation and attribute M why it is not so actually.

Chemical rockets keep losing weight by fuel burning ejection. Here I made a ideal case to get energy with (almost) zero loss of proper mass of spaceship.

[EDIT] Just to show how complex the exact velocity transformation is, say $v_n$, the speed of spaceship model n is given as
$$v_n=\frac{v_{n-1}+v}{1+\frac{v_{n-1}v}{c^2}}$$
where
$$v_0=v$$
Some examples
$$v_1=\frac{2v}{1+\frac{v^2}{c^2}}$$
$$v_2= \frac{\frac{2v}{1+\frac{v^2}{c^2}}+v}{1+\frac{\frac{2v}{1+\frac{v^2}{c^2}}v}{c^2}}$$
This transformation rule prohibits $v_n$ to exceed c.

 

[Omega0]

Instead of using relativistic mass, just use energy, and everything is fine.

yeah Hendrik, and if someone reached QM like you expert, it is definitely understood that an energy exchange rules the world - not forces. I really don't want to be arrogant, I just try to help somehow.

 

[Grasshopper]

Ehm , I know you have helped many people in these forums, I cannot judge my self the qualitative understanding of the OP (since I don't know relativity myself) but is it that bad, like playing random keys at piano? random relativity concepts and random words put together?

Pretty much bad in most every discipline. For example, go try to learn how to sing rock music, and then after years of frustration at not having vocal stamina, dealing with post-nasal drip, not being able to control your voice consistently on high or loud note, you’ll enroll in voice lessons. At that point, much time will have to be spent unlearning bad habits before proper technique training can really set in.

So it is with everything else in my experience. Unless you’re truly one of those who’ve been blessed by Odin, you’ll develop bad habits and have to unlearn them. And they’ll keep cropping up to mislead you for a long time after you THOUGHT you actually put them to bed. I hate to rant at you, but it really is much wiser to at least start on the path those who have succeeded before have taken, and to not leave it until a certain degree of competency has been reached.

Otherwise you’ll be up a creek.

 

[vanhees71]

The need of infinite energy to reach light speed can look a bit more intuitive if you notice that it also takes an infinite energy to stop something that has a nonzero mass and is moving towards you with speed $c$. Therefore, even a single electron can have effects similar to nuclear explosion or worse if it's moving fast enough (in Earth's reference frame) and collides with Earth. No energy is lost irreversibly when accelerating an object to higher and higher speeds.

That's self-contradictory. If "something" has a non-zero invariant mass it cannot move towards you with the speed of light (or a larger speed).

 

[dextercioby]

I wasn't trying to imply I am beyond knowing the math. I think you have interpreted this as some sort of arrogance on my part. I didn't give any context to that statement so I can see why you may take it that way. The reason I started out with that statement is because I won't understand the math. So where I was coming from is that I would prefer it if we kept the math out of the answers as I simply won't "get it".
[...]

But, as you can see from the discussion here, undestanding an answer to your question about possibility or impossibility of reaching "c" by sequential acceleration requires knowing high school mathematics. In other words, you shouldn't ask a question by phrasing "I want a qualitative/wordy explanation, I do not know the math to understand a complicated answer", but instead write: "please, in your explanations keep the mathematics at a high-school level". And people will try to provide with what you want. Simply setting "discussion level B/basic" from the thread options will not suffice, generally, even though it should.

 

[Sagittarius A-Star]

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. This does not require an infinite amount of energy.

A constant longitudinal force $F= m\alpha$ leads to a constant proper acceleration $\alpha=\gamma^3 a$ but to a decreasing coordinate-acceleration.

For the needed kinetic energy see:

With proper acceleration $\alpha=\gamma^3 a$ (follows from relativistic velocity addition) and if the force is in direction of movement:

relativistic kinetic energy = $\int F \cdot ds = \int m\alpha \cdot ds = m \int \gamma^3 a \cdot ds = m \int \gamma^3 \frac{dv}{dt} \cdot ds = m\int_0^v \gamma^3 v \cdot d v = mc^2(\gamma -1)$

 

[robphy]

Here's an energy-momentum diagram that could be helpful.
(Energy runs to the right.
A corresponding spacetime diagram (a position vs time graph) has time running to the right.)

A particle of mass m=1.0 unit has relativistic energy and momentum, as measured in this "lab frame",
(E,p) along the hyperbola of radius 1.0 (called the "mass-shell").

The slope gives the velocity of the particle in this frame. The dotted rays correspond to the speed of light, which are the asymptotes of the hyperbolas.

Suppose one can change the velocity of the particle without changing its [invariant rest-]mass.
Then the particle state (E,p) is advanced along the m=1.0 hyperbola, as shown.

Suppose that, in the particle frame, the impulses are discrete and are equally-sized.
On the diagram, equal-size impulses correspond to equal areas of the hyperbolic sectors.
(The sector area is proportional to the rapidity [Minkowski-angle] change along the arc.)

Note that each successive of the "equal-size impulses in the particle frame"
corresponds "in the lab frame"
to successively smaller LARGER increments in relativistic energy and in relativistic momentum,
as well as but with smaller increments in velocity, which is asymptotically approaching the speed of light,
which means that successively larger increments in relativistic energy and momentum in the lab-frame are needed to further increment the speed in the lab-frame.

(To see what it looks like in the particle frame,
reset (E,p) on the m=1.0 hyperbola back to v=0.
Note that the speed of light on the diagram is still represented by the asymptotes of the hyperbola.)

Although it may not be obvious,
the area of hyperbolic sector with sides along v=0.0c and v=1.0c
is infinite.
So, this particle of [invariant rest-]mass m cannot reach
the speed of light with a finite number of such impulses.

Here's my desmos file for the diagram https://www.desmos.com/calculator/pyhb6ua0j3

(For more information on
other ways to try to
get this particle to reach the speed of light (which involves a change in [invariant rest-]mass),
here is an older post
https://www.physicsforums.com/threads/massless-photon.900960/page-2#post-5842652 )
By the way, one question in your original post is unclear.
I have reformatted it to ask for clarification.

...So my question is, is it more accurate when we say
“it requires an infinite amount of energy to reach the speed of light”
to change that to
“it requires an infinite amount of energy to reach the speed of light”

Something is unclear.

 

[DaveC426913]

So my question is, is it more accurate when we say “it requires an infinite amount of energy to reach the speed of light”

Avoid talking about infinities. Talk about limits instead.

As speed approaches c, the energy required to accelerate further increases without bound.

 

[James Demers]

I'm far from an expert on relativity, so maybe I'm in a position to see where the original poster is "going wrong".
Sitting in your spaceship, you fire the engines, and you feel an acceleration. You then turn them off, and feel no acceleration. What makes you think you're going 10 mph faster? You have to look outside, and see the stars going by 10 mph faster than before. BUT ... what's an "hour"? Your clocks are slowing down. What's a "mile"? Your measuring devices aren't what they were, either. Mistake #1 is thinking that you can make the "same" measurements you made before accelerating. The one thing your on-board laboratory CAN measure consistently is the speed of light - and it will always be c. Everything in your ship (time, space, mass) bends (from the viewpoint of an outside observer) in a way that keeps c constant - and it's all undetectable to you, unless you look out the window. And when you do, according to your ship's instruments, you'll find that the stars are not going by 10 mph faster.

 

[PeroK]

I'm far from an expert on relativity, so maybe I'm in a position to see where the original poster is "going wrong".
Sitting in your spaceship, you fire the engines, and you feel an acceleration. You then turn them off, and feel no acceleration. Think about what's changed. Inside the ship, nothing at all. Ignoring the fuel gauge, there's nothing different.

This is correct. Nothing has changed physically; only your inertial rest frame. In a sense that is what acceleration does: changes your inertial frame.

The rest of your post was, sadly, not correct at all.

 

[Ibix]

Not only is your mass increasing

Mass doesn't increase, as noted several times in this thread.

your clocks are slowing down

This is misleading. The point is that there is no absolute sense in which the clocks are doing anything.

From the perspective of an observer who was initially at rest with respect to the rocket, it is true that your clocks tick slowly and out of sync and your rulers are length contracted. If you work out the details you'll understand why this observer understands that the rocket observer will measure light to travel at $c$ - the way the rulers and clocks misbehave gives the result. The rocket observer, on the other hand, sees themself at rest so sees nothing unusual in their equipment so is unsurprised to measure speed $c$ for light.

Everything in your ship (time, space, mass) bends (from the viewpoint of an outside observer)

Absolutely not. Everything here is in flat spacetime.

As your speed increases to a healthy fraction of c, the mass of your toothbrush gets to be a thousand kilos

No, as already noted, unless your source is over half a century old.

 

[Grasshopper]

Mass doesn’t increase with speed.Edit-

Why does this even exist?

$M(v) = \frac{m}{\sqrt{1 - \frac{v^2}{c^2}}}$

It has “normal” mass in the formula. What is the value of it other than an irrational need to keep momentum as $p=Mv$?

Or was there some additional motivation?

 

[votingmachine]

I wasn't trying to imply I am beyond knowing the math. I think you have interpreted this as some sort of arrogance on my part. I didn't give any context to that statement so I can see why you may take it that way. The reason I started out with that statement is because I won't understand the math. So where I was coming from is that I would prefer it if we kept the math out of the answers as I simply won't "get it".

Let me give an analogy of what I mean. When you are answering questions you can give a "high level" overview without going into the details, this is what I was asking for (if possible). As an example of what I mean - let's say I am programmer and you are not IT literate at all and ask me what my program does. I can explain to you what it does and go into detail without showing you the code (aka math formulas). Even if the program is complicated (like SR) I can explain how it works along the lines of "x does this etc, just trust me). You can still understand (maybe that is a strong word in this context, maybe "make sense of" is better way to put it) the program (SR or GR) and "get it", but to prove it I'd need to show you the code (maths) and YOU would need to understand the programming language (understand the equations) to verify it. What I am essentially saying is that right now I don't need the verification part (as I can't grasp it yet), I will just trust what you say as true, so adding the math only complicates it more without any benefit at this point. I understand that to fully grasp SR and GR I need to understand the math - but right now none of it will make any sense, but talking in plain English will...

I just wanted to clear that up in case I came across as some arrogant fool that thinks they don’t need to lower themself to the math.

Except you started out with some basic math. Your example is just using an "F=ma" equation.

And that equation is incorrect at high velocities. Your general question seems to me to be in effect ... since F=ma, why can't you just accelerate to light speed?

The answer is that F=ma is only an accurate approximation for lower speeds.

I might be missing your question, but that is how I read it.

 

[mucker]

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?

 

[DrGreg]

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?

Historically there have been two different definitions of mass, relativistic mass which depended on velocity and was observer-dependent, and invariant mass (or rest mass) which is the same for all observers and does not vary with velocity.

Nowadays almost all professional relativists use only invariant mass, and therefore just call it "mass".

For more details, see our FAQ: What is relativistic mass and why it is not used much?

 

[anuttarasammyak]

that - your mass does not increase as you accelerate?

Momentum of moving body is
$$p=m\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}v$$
You take it
$$p=[m\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}]v$$
mass increase with velocity and Newton physics formula p=Mv keep standing.
But we now know we should take it
$$p=m[\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}v]=mu$$
mass remain constant and p=mu holds instead of good old days p=mv formula.

You may perceive mathematical merit of the new way. In your way both mass and velocity changes according to velocity. It complicates the calculation. But in the new way only u changes with velocity keeping m a constant number. We can focus on u only in calculation.

For an example
$$v=2.4\times 10^8 m/s < c$$
corresponds
$$u=4.0\times 10^8 m/s > c$$
v has upper limit of c but u has no upper limit so it can explain infinite momentum with constant mass,

As for energy
$$E=m\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}c^2=[m\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}]c^2$$
You take mass changes and Einstein's wisdom of E=Mc^2 stands. But we should take it
$$E=m\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}c^2=\sqrt{m^2c^4+p^2c^2}$$
with constant m and above explained p including constant m within. Formula E=mc^2 stands only for p=0 or the body is at rest. This is an exact reading of E=mc^2 formula.

So now increasing momentum and energy with velocity are interpreted not by increasing mass but increasing u, a new type of velocity which replace old v.

 

[Chris S]

Layperson's question: Is it true that an object with mass can never travel as fast as an object that has no mass? If true, does this fact answer the question?

 

[jbriggs444]

Layperson's question: Is it true that an object with mass can never travel as fast as an object that has no mass? If true, does this fact answer the question?

It is a fact, yes. But the question I see asked in the original post was "why does this argument that we can incrementally accelerate an object faster than the speed of light fail?"

Simply asserting that the argument reaches a false conclusion is unsatisfying. @mucker was already aware of that.

 

[RandyD123]

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

 

[Dale]

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

No. There is no such thing as “relative to a photon”, and we are not moving at c relative to a neutrino.

 

[Delta2]

There is no such thing as “relative to a photon”

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

 

[Dale]

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

Because any reference frame (tetrad) by definition has one timelike and three spacelike vectors, and so none of their integral curves can form a lightlike worldline. Furthermore, light travels at c in all inertial frames (tetrads and coordinates) so it cannot be at rest in one.

 

[PeterDonis]

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

See this relativity forum FAQ:

https://www.physicsforums.com/threads/rest-frame-of-a-photon.511170/

 

[Delta2]

Because any reference frame (tetrad) by definition has one timelike and three spacelike vectors, and so none of their integral curves can form a lightlike worldline. Furthermore, light travels at c in all inertial frames (tetrads and coordinates) so it cannot be at rest in one.

What do you mean by "their integral curves" and a "lightlike wordline"? The worldline of a photon?

 

[anuttarasammyak]

All the IFRs share the light cone. In photon rest frame, if there is, the light cone should shrink to a point there. I think it is serious change of spacetime structure from IFR.

 

[PeterDonis]

What do you mean by "their integral curves" and a "lightlike wordline"? The worldline of a photon?

If @Dale's explanation is too complicated (and it might be for a B-level thread), please read the FAQ article I linked to, which is much simpler.

 

[PeterDonis]

In photon rest frame, if there is, the light cone should shrink to a point there.

This is incorrect as an explanation of why there cannot be a rest frame for a photon. The correct simple explanation is in the FAQ article I linked to. More complicated explanations are probably beyond the scope of a B-level thread.

 

[Dale]

What do you mean by "their integral curves" and a "lightlike wordline"? The worldline of a photon?

A lightlike worldline is the worldline of a classical flash of light. A photon doesn’t actually have a worldline or even a position. I wish people would not use the word “photon” outside of actual discussions of QM photons

 

[PeterDonis]

I wish people would not use the word “photon” outside of actual discussions of QM photons

I agree with this (and I would note that the FAQ article I linked to earlier probably needs its title and some of its text edited to address this).

 

[James Demers]

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?

"Relativistic mass" has largely been abandoned because it's not a particularly useful concept.

"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. This does not require an infinite amount of energy."

Repeated additions of energy do cause repeated increases in momentum ... but relativity teaches us that this does NOT mean repeated additions to velocity. That's the mistake you're making. You cannot reach c, and saying that "infinite energy" will get you there is meaningless. (Converting the entire mass of the universe to propulsive energy still won't get you - or even a single electron - to c.)

 

[Ibix]

So one thing I'd just like to clarify (as everyone keeps saying) is that - your mass does not increase as you accelerate?

As DrGreg says, there are two just about plausible generalisations of Newtonian "mass" to relativistic physics, relativistic mass and invariant (aka rest) mass. The relativistic mass of something moving with respect to you increases, the invariant mass does not.

Relativistic mass turns into a mess. As anuttarasamyak says, it makes Einstein's momentum formula look the same as Newton's. But making your better theory look like your poorer one is kinda backwards. And you find you end up having to define two more quantities, longitudinal relativistic mass and transverse relativistic mass, to pull a similar trick with $F=ma$, and now we've got three different definitions of mass for different applications and we're getting into silly territory. Invariant mass has none of these problems, fits better into a modern understanding of relativity which is all about invariants, and fits well with general relativity where the basis of the definition of relativistic mass is rather dubious.

Most professionals now never use relativistic mass at all, and if they say "mass" they mean invariant mass. And a physicist by the name of Lev Okun, @levokun, made a serious push in the 90s/2000s to get relativistic mass stamped out of physics teaching altogether because it led to such confusion.

Popsci has never quite caught on to this and will still talk about mass increasing with speed.