[Pivz_ (OP)]

I would like to start off with saying that I know there are other reasons why it's not possible to reach c, however I often hear the argument that the speed of light cannot be reched by a particle with mass because as its velocity increases so does the mass... (blah-blah-blah - I'm not going to explain the whole argument in detail because you probably you know how it goes) ...therefore it would take an infinite amount of energy to reach the speed of light.
It is true that for a stationary observer the mass of a particle (from now on it's going to be a spaceship, because they are awesome) increases as it accelerates. But in the frame of reference of the ship (as seen by, for example, passengers on board) the mass of the ship is equal to its rest mass - no relativistic mass is added. In fact, for them, the opposite happens - the mass of a particle, they were stationary to before they started moving (from now on it will be called "the Earth"), is now increasing. In the end both the passengers on the ship and [petty] Earthlings think of each other as getting heavier.
If, in the frame reference of the ship, it's of the same mass as when it started, then this argument is false, right? Or maybe my logic is flawed? Any thoughts for either side?
Thanks
PS sorry for any grammar mistakes - I'm not a native speaker

[PmbPhy]

...because as its velocity increases so does the mass... (blah-blah-blah - I'm not going to explain the whole argument in detail because you probably you know how it goes) ...therefore it would take an infinite amount of energy to reach the speed of light.

That part is correct but as you continue, your writing gets more confusing. For example, you write

It is true that for a stationary observer the mass of a particle (from now on it's going to be a spaceship, because they are awesome) increases as it accelerates.

What is that supposed to mean? The mass of a particle as measured from the rest frame of the particle is called the particle's proper mass. What is this "stationary observer" that you mentioned here? Is it a frame of reference in which the particle is moving?

Observers at rest with respect to the rocket ship (or particle) will measure it to have the same mass it had before it started to move.

Correct.

But in the frame of reference of the ship (as seen by, for example, passengers on board) the mass of the ship is equal to its rest mass - no relativistic mass is added. In fact, for them, the opposite happens - the mass of a particle, they were stationary to before they started moving (from now on it will be called "the Earth"), is now increasing.

I assume that you're talking about the following scenario:

For t < 0 a rocket ship and a particle is at rest in the inertial frame of reference S. When t = 0 the rocket ship starts to accelerate while the particle remains at rest. For observers at rest in frame S the particle's mass remains constant having the value of its proper mass. The mass of the rocket as observed from this frame is increasing. From the rocket's frame of reference S' the rocket's proper mass remains constant (disregarding fuel ejected to create thrust. So by "rocket" we'll mean "command capsule" where astronauts live) but observers in S' will determine that the mass of the rocket is constant but that the particle's mass is increasing.

Note: This is tricky because when we say "the mass of the particle as measured by observers in frame S' " we're talking about a non-accelerating frame that is instantaneously moving at the same speed as S'. This is because the gravitational potential contributes to the relativistic mass of a particle moving in a gravitational field and an accelerating frame is equivalent to a gravitational field. For more on this please see my website on this at: http://home.comcast.net/~peter.m.brown/gr/uniform_force.htm

See Eqs. 9 and 12

[evan_au]

in the frame of reference of the ship, it's of the same mass as when it started

This is true - observers on the ship will not notice a change in mass of the command capsule.

However, from the viewpoint of the observers on the ship, light still passes them at speed c, so despite burning lots of fuel, they have made no progress towards their goal of traveling at speed c.

So they will need infinitely more fuel than they have already burned to reach speed c.

Ideally, they would know this before they set off, so they would set themselves a more realistic goal, like reaching Alpha Centauri...

[Pivz_ (OP)]

For t < 0 a rocket ship and a particle is at rest in the inertial frame of reference S. When t = 0 the rocket ship starts to accelerate while the particle remains at rest. For observers at rest in frame S the particle's mass remains constant having the value of its proper mass. The mass of the rocket as observed from this frame is increasing. From the rocket's frame of reference S' the rocket's proper mass remains constant (disregarding fuel ejected to create thrust. So by "rocket" we'll mean "command capsule" where astronauts live) but observers in S' will determine that the mass of the rocket is constant but that the particle's mass is increasing.

That's exactly what I meant.

For more on this please see my website on this at: newbielink:http://home.comcast.net/~peter.m.brown/gr/uniform_force.htm [nonactive]

That's an awesome looking website. Maybe someone with more than just AS physics knowledge would understand things described there, because I don't []

However, from the viewpoint of the observers on the ship, light still passes them at speed c, so despite burning lots of fuel, they have made no progress towards their goal of traveling at speed c.

So they will need infinitely more fuel than they have already burned to reach speed c.

So is it possible for two objects to travel at/faster than c relative to each other within the observable universe? (and is there a way for these objects to communicate with each other?). If that's the case then how will this work if all observers must agree on the speed of light? Will they disagree on the location of a photon?

[evan_au]

So is it possible for two objects to travel at/faster than c relative to each other within the observable universe?

No.

I think that many questions like this come down to the addition of relativistic velocities, which is different from just adding up velocities (as Galileo & Newton would have done).

So if an observer can see one object moving away from him at half the speed of light in one direction, and another object moving away from him at half the speed of light in the opposite direction, you might expect that the two "end" observers would be moving away from each other at the speed of light.

However, when you apply the relativistic addition of velocity formula, you find that they are only moving away from each other at 80% of the speed of light, so they can still communicate with each other.

[PmbPhy]

So is it possible for two objects to travel at/faster than c relative to each other ...

What do you mean by "relative to each other"?  If, in frame S, a particle is moving at 0.9c to the left and another particle is moving at 0.9c to the right then the particles are moving away from each other at 1.8c mph as measured in frame S where by "moving away from each other" I mean that the time rate of increase of the distance between the two particles is 1.8c. However frame the rest frame of either particle the speed of the other particle is less than the speed of light.

Please describe what you mean by these two particles moving "relative to each other".