# Questioning examples often used to introduce the basics of relativity

Hi all,

I've been dusting off the grey matter a bit, and have a question regarding an example I found used in both high school and university physics when broaching the topic of Special Relativity.

I do tend to ramble, and so I'll point you to my blog post at sumofprimes.blogspot.com/2008/04/relativistic-mass-in-introductory.html if you care to read the whole spiel. For those that don't wish to suffer, here it is more briefly stated:

Regardless of any other reasons that may prevent an object with mass from accelerating to the speed of light, the often-used argument that an infinite amount of fuel would be required seems inherently flawed -- this argument typically presents the apparent mass increase of an object that's accelerating (as observed from a stationary frame of reference), then proceeds with the reasoning that if the apparent mass is approaching infinity, then the amount of "fuel" required to continue acceleration would approach infinity, too. The main flaw with this line of reasoning appears to be that:

a.) if we assume that the object is carrying the fuel with it for the trip, then
b.) the observed mass of the carried fuel will increase proportionately with the increase in the mass of the accelerating object (and so, too, approach infinity)

So -- the static observer would see the mass of the system as a whole increase (m[observed] = m[rest]/(1-(v^2/c^2))^0.5), whereas the "traveler" would simply see "Ah, yup, 29,875kg of fuel left now..." -- a steadily decreasing amount of fuel as it was consumed.

I think I may have expanded on this and a few other niceties more effectively in the linked blog post, if you have the time to read it.

Is my reasoning/example flawed, or is this really a poor argument when indicating why accelerating to the speed of light is problematic?

- Sum

russ_watters
Mentor
Well, just because two numbers are approaching infinity, doesn't make them equal. Ie, the expression 2x/x (as x->infinity) doesn't approach 1 at high values for X, it still approaches 2. So this changes nothing.

Well, just because two numbers are approaching infinity, doesn't make them equal. Ie, the expression 2x/x (as x->infinity) doesn't approach 1 at high values for X, it still approaches 2. So this changes nothing.

Hi Russ,

Thank you for the speedy reply.

I agree with your statement, however it doesn't appear to relate to the example provided -- when considering the equation m1=m0/(1-(v^2/c^2))^0.5 for calculating the relativistic mass, we can see that the denominator is not influenced by the masses in question, only the velocity. As such, the degree of the increase in the apparent/relativistic mass of the "craft" and its "fuel" will be proportionate, and the mass[craft]:mass[propellant] ratio will remain constant when considering velocity (the fuel will of course diminish as it's consumed, but the relativistic increase in mass will remain proportionate to the ship's mass as v increases).

- Sum

Thank you to whoever commented on my blog! The comment consisted of "This treats Relativistic mass as if it was a real changing thing.
Science does not treat mass that way, nor did Einstein.
Better to think of the formula needing to factor both the mass and the speed by the Gamma Factor to give a result and not that any “Mass” is actually changing form any perspective.
Only that different observers must use different Gamma Factors.
Think of “relativistic mass” as an abstraction that can be used in the math."

Which seems to match my current understanding of things. As in my response to the comment, the author appears to be in agreement that this is a bad example to present when explaining relativity. They appear to have refuted the example with the provision of how things should actually be considered, whereas my attempt to refute it is by trying to expose a logical fallacy ("Well, if _that's_ how things work, then _this_ should be the result, not what you're proposing with this example of infinite fuel")

- Sum

Again for the anonymous poster (or anyone else, for that matter), what's your opinion on my commentary about the time dilation side of things for the travels of that hypothetical vessel?

1. Only taking into account the time dilation aspect of things, is my description accurate for why an observer in a stationary frame of reference would never see an object with constant acceleration (from the object's frame of reference) reach light speed?

2. Again from the accelerating object's frame of reference, what reading would you recommend relating to the physics of an object whose velocity is approaching c?

Last edited:
Please bear with me as I continue with this thought experiment, and set me on the right path if I head off in the wrong direction.

The "infinite energy" requirement for an object with mass to achieve light speed does appear to stand. It does appear to have a couple of conditions, though:
- 1.) it would apply in order for such a velocity to be achieved in a time that could be seen from a stationary frame of reference
- 2.) it seems to apply not from the commonly used relativistic mass standpoint, but rather from a time dilation standpoint.

Following the time dilation tangent:
- In relation to 1., above, if an object had an acceleration (from its frame of reference) that was constant or increasing in a linear fashion, the time dilation effect for the observer would outpace the increase in velocity for the object, and so for v to equal c would take an infinite amount of time;
- In relation to 2. and the above statement, for v to equal or surpass c in a finite amount of time from that stationary frame of reference would require that the object's acceleration increase geometrically, at a rate which equalled or surpassed the time dilation effect...without working out the details, and at first glance, as v approached c, the acceleration of the object would seem to have to approach infinity in order for the observed increase in v to reach or surpass c from that stationary frame's standpoint in a finite amount of time

Last edited:
Regardless of any other reasons that may prevent an object with mass from accelerating to the speed of light, the often-used argument that an infinite amount of fuel would be required seems inherently flawed -- this argument typically presents the apparent mass increase of an object that's accelerating (as observed from a stationary frame of reference), then proceeds with the reasoning that if the apparent mass is approaching infinity, then the amount of "fuel" required to continue acceleration would approach infinity, too. The main flaw with this line of reasoning appears to be that:

a.) if we assume that the object is carrying the fuel with it for the trip, then
b.) the observed mass of the carried fuel will increase proportionately with the increase in the mass of the accelerating object (and so, too, approach infinity)

The problem with this explanation is that, by itself, relativistic mass does not prevent the observation of superluminal objects. For it does not imply the addition of velocity formula and therefore two observers in two inertial reference frames moving in opposite directions may observe the other to be traveling at a speed greater than c. Thus it is not a sufficient reason for the existence of a fundamental speed limit for any object.

Last edited:
Dale
Mentor
2021 Award
my attempt to refute it is by trying to expose a logical fallacy
I didn't see any logical fallacy. It is well-known that relativistic mass is not invariant, so the disagreement about the relativistic mass does not present a logical fallacy.

Was there something else you were refering to?

Again for the anonymous poster (or anyone else, for that matter), what's your opinion on my commentary about the time dilation side of things for the travels of that hypothetical vessel?

1. Only taking into account the time dilation aspect of things, is my description accurate for why an observer in a stationary frame of reference would never see an object with constant acceleration (from the object's frame of reference) reach light speed?

2. Again from the accelerating object's frame of reference, what reading would you recommend relating to the physics of an object whose velocity is approaching c?

One difficulty is that you can not even approach the speed of light. If you have a velocity of say 0.99c according to one observer, he might assume you are one percent short of the speed of light, but to a co-moving observer you are stationary and still 100% short of the speed of light.

Another factor is that the equations for the relativistic acceleration of a rocket show that a rocket with constant proper acceleration will take an infinite period of time to reach the speed of light with any constant finite acceleration. See this FAQ http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html [Broken]

Now lets say you have a finite amount of fuel, for example 1 tonne and your rocket engine ejects the fuel at the rate of 1kg per hour. (You have to eject mass in order to accelerate.) Your fuel would be ejected after 1000 hours of proper time. Now even though 1000 hrs of proper time aboard the rocket equates to a much longer time in the frame of an observer that is not accelerating and was at rest with rocket prior to it launching, the time for fuel to run out is finite for any observer despite considerations of the fuel increasing due to relativistic mass considerations.

Another consideration is how stored energy transforms. For example does one tonne of nuclear fuel stored onboard a rocket with a velocity of 0.8c really have 1.666 times the stored energy of the rest energy of the fuel that is usable to accelerate the rocket? This is a question of "internal energy" and the exact answer is not completely clear and relates to question of thermodynamics in relativity. That is a subject that does not appear to be completely resolved at this time.

Another example:
Say we have a cylinder of compressed gas that propels a projectile to 0.6c when the pressure is released. This is a simple rocket propellant device. Now say the cylinder is onboard a rocket that is moving at 0.6c according to one observer. If the cylinder is pointing to the rear of the rocket and the gas is released, then the projectile will de-accelerated from 0.6c to 0.0c according to that observer, by simple use of the relativistic velociy addition equation. (This is also related to long discussion in another thread on the invariance or otherwise of stored pressure.) From this simple example it can be seen that the effective (useable) stored energy of the cylinder of gas is no different in the primed or unprimed frames as far as its use as a propellant is concerned. So in this example the increased relativistic mass of the propellant has not translated to increased useable propellant energy and to be consistant this principle probably applies to any form of stored energy.

Last edited by a moderator:
One difficulty is that you can not even approach the speed of light. If you have a velocity of say 0.99c according to one observer, he might assume you are one percent short of the speed of light, but to a co-moving observer you are stationary and still 100% short of the speed of light.

Another factor is that the equations for the relativistic acceleration of a rocket show that a rocket with constant proper acceleration will take an infinite period of time to reach the speed of light with any constant finite acceleration. See this FAQ math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

Now lets say you have a finite amount of fuel, for example 1 tonne and your rocket engine ejects the fuel at the rate of 1kg per hour. (You have to eject mass in order to accelerate.) Your fuel would be ejected after 1000 hours of proper time. Now even though 1000 hrs of proper time aboard the rocket equates to a much longer time in the frame of an observer that is not accelerating and was at rest with rocket prior to it launching, the time for fuel to run out is finite for any observer despite considerations of the fuel increasing due to relativistic mass considerations.

Another consideration is how stored energy transforms. For example does one tonne of nuclear fuel stored onboard a rocket with a velocity of 0.8c really have 1.666 times the stored energy of the rest energy of the fuel that is usable to accelerate the rocket? This is a question of "internal energy" and the exact answer is not completely clear and relates to question of thermodynamics in relativity. That is a subject that does not appear to be completely resolved at this time.

Another example:
Say we have a cylinder of compressed gas that propels a projectile to 0.6c when the pressure is released. This is a simple rocket propellant device. Now say the cylinder is onboard a rocket that is moving at 0.6c according to one observer. If the cylinder is pointing to the rear of the rocket and the gas is released, then the projectile will de-accelerated from 0.6c to 0.0c according to that observer, by simple use of the relativistic velociy addition equation. (This is also related to long discussion in another thread on the invariance or otherwise of stored pressure.) From this simple example it can be seen that the effective (useable) stored energy of the cylinder of gas is no different in the primed or unprimed frames as far as its use as a propellant is concerned. So in this example the increased relativistic mass of the propellant has not translated to increased useable propellant energy and to be consistant this principle probably applies to any form of stored energy.

Ah, thank you very much! The constant acceleration equating to a rocket taking an infinite amount of time to reach c was exactly what I was trying to convey in post regarding time dilation :) (Oh, to be able to express myself more clearly ;) At the least, it's good to see that reasoning hasn't failed me on that topic, even if expression has.

@DaleSpam -- the logical fallacy I was attempting to establish was:
- an argument is posed saying "a rocket cannot reach light speed, since its relativistic mass will approach infinity, and so require an infinite amount of energy to continue accelerating"
- the argument as presented didn't appear to be logical to me: it's arguing that the rocket's relativistic mass will approach infinity, but treating the fuel it carries as though it wouldn't experience the same increase, whereas if it's carried with the rocket, its mass should approach infinity, too
[In this light, I was trying to establish that I agree that accelerating to c is a problem, but that I'm more inclined to go with the time dilation aspect of things to indicate why, rather than the relativistic mass example commonly used]

NateTG
Homework Helper
@DaleSpam -- the logical fallacy I was attempting to establish was:
- an argument is posed saying "a rocket cannot reach light speed, since its relativistic mass will approach infinity, and so require an infinite amount of energy to continue accelerating"
- the argument as presented didn't appear to be logical to me: it's arguing that the rocket's relativistic mass will approach infinity, but treating the fuel it carries as though it wouldn't experience the same increase, whereas if it's carried with the rocket, its mass should approach infinity, too
[In this light, I was trying to establish that I agree that accelerating to c is a problem, but that I'm more inclined to go with the time dilation aspect of things to indicate why, rather than the relativistic mass example commonly used]

It's pretty easy to work with momentum:
Consider a rocket with laser drive that propels itself by shooting as much light out the back as you like, and assume that the rocket has a vehicle mass of $m$ and a fuel mass of $M$. The rocket has a specific impulse of at most $c$ so we can solve the momentum equation:
$$p=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}=Mc$$
$$v=\pm c\sqrt{\frac{M^2}{m^2+M^2}}$$
And, the absolute value of the square root is clearly less than 1 so
$$|v|<c$$

Last edited:
It's pretty easy to work with momentum:
Consider a rocket with laser drive that propels itself by shooting as much light out the back as you like, and assume that the rocket has a vehicle mass of $m$ and a fuel mass of $M$. The rocket has a specific impulse of at most $c$ so we can solve the momentum equation:
$$p=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}=Mc$$
$$v=\pm c\sqrt{\frac{M^2}{m^2+M^2}}$$
And, the absolute value of the square root is clearly less than 1 so
$$|v|<c$$

The OP is proposing that the fuel mass also increases by gamma to give:

$$p=\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}=\frac{Mc}{\sqrt{1-\frac{v^2}{c^2}}}$$
$$v= c\frac{M}{m}$$

which still gives $$|v|<c$$ when fuel mass M is less than total rocket mass m and where m = M + mass of the unfueled rocket.

Dale
Mentor
2021 Award
@DaleSpam -- the logical fallacy I was attempting to establish was:
- an argument is posed saying "a rocket cannot reach light speed, since its relativistic mass will approach infinity, and so require an infinite amount of energy to continue accelerating"
- the argument as presented didn't appear to be logical to me: it's arguing that the rocket's relativistic mass will approach infinity, but treating the fuel it carries as though it wouldn't experience the same increase, whereas if it's carried with the rocket, its mass should approach infinity, too
[In this light, I was trying to establish that I agree that accelerating to c is a problem, but that I'm more inclined to go with the time dilation aspect of things to indicate why, rather than the relativistic mass example commonly used]
Hi Sum, Nate and Kev already gave good explanations using the relativistic mass concept and conservation of momentum. You can also derive relativistic versions of the standard rocket equations to determine things like the amount of fuel required to accelerate a specific payload to relativistic final velocity. The basic point is that even though the relativistic mass of the fuel increases when you actually do the calculations you still don't accelerate to c for any finite amout of fuel.

To sumofprimes;
Energy is transferred at c, light speed. The faster an object moves, the longer it takes for this transfer. The extended time is interpreted as inertial resistance (increase in mass).

Hi all,

Thanks for the replies! Again, I agree that for the observer in a static frame of reference will never see the traveler reach c, for the reasons we've covered.

Even though the stationary observer will never see the traveler reach c, strictly from the viewpoint of the traveler, what can they be expected to experience?

We can throw out some purely hypothetical numbers (a.k.a. unrealistic, but should do for examples), if you don't mind, of course.

Let's say you are a traveler on a craft that's carrying 200 units of fuel. Let's also say that at low velocities, burning one unit of fuel over the course of a day will result in a change of velocity of 0.01 c. What state will you observe yourself to be in after going through half of your fuel? All of your fuel? [If desired, we'll throw in that the craft has an arbitrary rest mass...say, 100 units...or 1000 units, if it better suits you]

Hang on...
...
...<revisits momemtum equations>...
...<waits for them to sink in>...
...
...ok, that makes sense -- please disregard the question, above, and thank you!

There is still something nagging at the back of my mind on the matter, but I'll wait until it's formed into something coherent so that I can pose a well formed question -- as you can see from the evolution of the thread, what I ask in type doesn't necessarily reflect the question I'm actually trying to ask (thank you for your patience!).

Hi all,

Thanks for the replies! Again, I agree that for the observer in a static frame of reference will never see the traveler reach c, for the reasons we've covered.

Even though the stationary observer will never see the traveler reach c, strictly from the viewpoint of the traveler, what can they be expected to experience?

We can throw out some purely hypothetical numbers (a.k.a. unrealistic, but should do for examples), if you don't mind, of course.

Let's say you are a traveler on a craft that's carrying 200 units of fuel. Let's also say that at low velocities, burning one unit of fuel over the course of a day will result in a change of velocity of 0.01 c. What state will you observe yourself to be in after going through half of your fuel? All of your fuel? [If desired, we'll throw in that the craft has an arbitrary rest mass...say, 100 units...or 1000 units, if it better suits you]

Hang on...
...
...<revisits momemtum equations>...
...<waits for them to sink in>...
...
...ok, that makes sense -- please disregard the question, above, and thank you!

There is still something nagging at the back of my mind on the matter, but I'll wait until it's formed into something coherent so that I can pose a well formed question -- as you can see from the evolution of the thread, what I ask in type doesn't necessarily reflect the question I'm actually trying to ask (thank you for your patience!).

The relativistic mass, by itself, does not prevent the observation of superluminal objects. For it does not imply the addition of velocity formula and therefore two observers in two inertial reference frames moving in opposite directions may observe the other to be traveling at a speed greater than c. Thus it is not a sufficient reason for the existence of a fundamental speed limit for any object

NateTG
Homework Helper
As a note: if you look at the solved momentum equation:
$$|v| \leq c \sqrt{\frac{M^2}{M^2+m^2}}$$

It should be clear that the square root is going to be less than one regardless of mass-scaling.

If a rocket accelerates half the remaining difference to the speed of light every second, then it's constantly accelerating, and will have average speed c in the limit, but it won't hit c in any finite time.

The relativistic mass, by itself, does not prevent the observation of superluminal objects. For it does not imply the addition of velocity formula and therefore two observers in two inertial reference frames moving in opposite directions may observe the other to be traveling at a speed greater than c. Thus it is not a sufficient reason for the existence of a fundamental speed limit for any object

Thanks Bosemann, that's one of the things that was nagging at me ;)