# Length contraction speed of light

The Special Theory of Relativity tells us that a moving object eg spaceship measures shorter in its direction of motion as its velocity increases.

At the speed of light it would have zero length, but infinite mass.

1. would the spaceship disappear from the perspective of an observer outside the ship at the speed of light, having zero length? where would it go?

2. at the speed of light, with zero length, what would a person inside the spaceship perceive?

3. if the object has zero length how can it have infinite mass?

Can the above occur in reality or would it lead to paradoxical results?

• Welcome to PSE, it can't achieve the speed of light, so there is no paradox. Also, I would Google regarding how inertial mass is treated in modern physics, the idea that mass increases with velocity is not straightforward. en.wikipedia.org/wiki/… – user179430 Dec 24 '17 at 0:41
• This question does not show any research effort (in particular, research here for the numerous nearly identical questions). – Alfred Centauri Dec 24 '17 at 1:51
• "at the speed of light, with zero length, what would a person inside the spaceship perceive?" - as has been mentioned here almost too many times to count, there is no inertial frame of reference with relative speed $c$, i.e., anything at rest with respect to itself does not have speed $c$ relative to any inertial reference frame. Think about that! – Alfred Centauri Dec 24 '17 at 1:55

Although mathematically allowed, the limiting case where the massive object reaches the speed of light is not practically realizable. This is often the situation with scenarios found in special and general relativity. There are limits that can never be reached in the physical world. A massive object reaching the speed of light is one such case.

Also remember the length contraction and increase in mass are relative observations. For the person inside the space ship, the length and mass remains the same. It's only the external observer that looks at the space ship speeding past it (i.e., a transverse observer as Steve explained) that will observe the length contraction.

• Oh dear, corrected it. – flippiefanus Dec 25 '17 at 5:44
• How is $v=c$ mathematically allowed for a body with nonzero mass? No Lorentz transformation boosts a timelike interval to a lightlike interval. – PM 2Ring Jun 21 at 9:09

I asked a physicist this same question, and the question was answered in a straight forward fashion, which given my novice understanding of physics is what I had been seeking:

The most important thing to remember here is that you can’t actually get to the speed of light. More than merely hypothetical, this is impossible. 1) applying the equations directly, you get a spaceship with zero length but the usual height and width. 2) at every speed, things appear normal on board the spaceship. You never perceive yourself as moving. 3) it’s not just an issue with being very thin; nothing can have infinite mass.

Let's forget about what's possible physically, it doesn't matter for this discussion.

The "issue" is that even mathematically there is no rest frame for a lightspeed particle. In relativity, the main point is that it doesn't matter who's moving: If I'm moving for an observer on earth with velocity $$v$$, then for me, the earth is moving with velocity $$v$$ (talking about magnitude, not direction). But if you try to "sit on a photon" and look around, you get into trouble (the whole universe would need to travel with lightspeed from perspective of a photon). Still, you can make sense of it if you take the limit carefully.

Here it goes:

A lightspeed observer perceives no time passing from beginning of their journey to the end. The universe is also infinitely contracted, it's a 2D pancake, there is no distance to travel, everything is here. That means, that the beginning and end of your journey are the same point in space and the same time. It's basically teleportation for you.

When a photon is emitted from your light source, it "travels" to the screen when it is absorbed (or reflected, re-emitted, however you want to see). But the photon doesn't live through that. He's born and dies at the same time, it didn't age or feel time, and with regard to it, the light source, the screen, and every point in between, are one and the same point, where he exists for that single moment in time.

If you turn this around to the external observer: you can never see a particle traveling at light speed, without intercepting it and effectively stopping its journey, because it has no time to emit anything, so it can't do anything on its own without hitting another particle (scattering). You've never seen light travelling at lightspeed, you can only see its arrival.

Taking the limit again: if you consider a very fast, but not quite lightspeed particle, whatever it emits, will be relativistically beamed in the forward direction (angle aberration and red/blueshift), so that it's redshifted to almost nothing in backward direction and blueshifted + aberrated in the forward direction so that it arrives barely before the particle does. In the limit, it would all arrive at the same time in the same direction, effectively meaning that it all just travels together (so nothing happened, there's no difference if it splits apart or if it doesn't, it's all just one chunk).

Mathematically speaking, you have to consider a space-time distance between two events in Minkowski metric, which is $$s^2=(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2-(ct_1-ct_2)^2$$. This is zero if you stay on the light cone (points in space-time that are connected with a light ray), which is a theoretical restatement of "for photon, the beginning and the end are the same".

To think about it another, stranger way: light circumvents the concept of distance and takes shortcuts through time and space, connecting distant events as if there was no distance at all, but can only do so for events, that for an external observer obey the distance=lightspeed×time connection. This makes the space-time strangely connected with specific rules: when you send light signals to some other place, the information from perspective of a photon instantaneously teleports in space and time to the final destination, but because it also teleported in time (relative to our external observer coordinates, which are not squished into a pancake), we must still wait. In curved time-space, for example, because of massive objects, these "shortcuts" are not straight, but they are still "instantaneous" for light - it's just criss-crossed in some curved manner so you can't intuitively tell which space-time point light will "teleport" to by simply drawing a straight line. Notice that in curved space-time light can get stretched (redshifted) during flight. That seems paradoxical at first, because nothing can happen to light when it's travelling, because it feels no time. However, there is no paradox, the light is the same, it still just went "poof" from source to the final observer, the catch is, that the observer is "playing it back" at a different rate than the source "played it", because their times don't run at the same rate.

The infinite mass thing is just a "helper" from non-relativistic mechanics, it actually isn't a sensible thing to say. Mostly, mass is what we call "rest mass", and thus is constant. Energy $$E=\sqrt{(mc^2)^2+p^2}$$ is the one that depends on velocity ($$p$$ is momentum), and if your mass is nonzero, you would need infinite energy to reach speed of light. Note that the mass $$m$$ doesn't change, only the momentum does, when you accelerate.

• "when you send light signals to some other place, the information instantaneouslyteleports in space and time to the final destination, but because it also teleported in time, we must still wait" I kind of get what you're trying to say, but I think that it's probably more misleading than helpful. – PM 2Ring Jun 21 at 9:17
• Added an important distinction in bold, I hope it helsp. – orion Jun 21 at 9:29
• Ok, but talking about the "perspective of the photon" implies that a photon has a rest frame. But that's simply not true, as discussed here. Ben Crowell discusses the history of this important idea here. As Ben says, Einstein realised that if a photon had a rest frame we'd be able to see frozen electromagnetic waves, which contradicts classical electromagnetic theory and observations; that realisation led to the special theory of relativity. – PM 2Ring Jun 21 at 10:14
• Exactly, there's no such frame, that's my first paragraph above; that's why you only have a single "point" and no time and space. I'm trying to convey the physical content of this lack of rest frame. We can say there's no such thing, but by looking at the limits and the way dimensions become degenerate, you also learn why and maybe understand space-time a bit better. – orion Jun 21 at 11:35