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96

Suppose we take the spacetime point of your conception as the origin, $(t=0, x=0)$, then the spacetime point for your birth would be $(t=T, x=uT)$. The time $T$ is approximately $9$ months, and we are writing the spatial position of your birth as $x=uT$ where $u$ is a velocity. The velocity $u$ can be any value from zero (i.e. born in the same spot as ...

73

The speed $c$ that is constant is so when measured locally relative to a freefalling frame (i.e. one for which all points follow spacefime geodesics wrt to the metric $g$). Local means that the frame's extent must be "small" enough that it can be thought of as flat: think of this as zooming in on the spacetime manifold, which is a smooth object, with enough ...

65

Let's suppress some dimensions to simplify: $$\Delta s^2 = -(c\Delta t)^2 + \Delta x^2$$ This quantity $$\Delta s^2$$ is preserved by changes of reference frame, just as in Galilean physics the quantity $$\Delta r^2 = \Delta x^2 + \Delta y^2$$ is preserved by rotations. Notice it is also the equation of a hyperbola. Thus, the effect of a frame shift is ...

45

No, you can't, for a couple of different reasons. The first is the difficulty of folding paper more than a few times. Mythbusters managed to fold one sheet 11 times I think, using a very large sheet of paper and the help of a steamroller. It took a lot longer than 5 seconds per fold. The second issue is more fundamental. You could resolve the first issue ...

42

No two things in the universe happen "instantaneously", unless they are at exactly the same location, because "instantaneously" would have different meanings for observers moving at different velocities. Maxwell's equations, which describe electromagnetic interactions perfectly for most practical purposes, contain time-dependent terms that describe the ...

42

It is both. Or even indeterminate. It is important to note that $\Sigma \vec F = m \vec a$ does not express a cause-effect relationship. Causes always preceed effects, so a causal relationship is given by an equation of the form $f(t)=g(t_r)$ where $t_r<t$ or more commonly $f(t)=\int_{-\infty}^{t} g(t_r) \ dt_r$ and $t_r$ is called the retarded time. ...

40

The article you linked is about a false alarm that was caused by some error in data processing. The neutrinos didn't go faster than light. The popular media of course blew this out of proportion, as usual. It's not really like a speed limit/barrier. It's the geometry of space. It's not like a highway sign - it's more like the fact that you can't go more ...

34

This was a reference to the apparent measurement that neutrinos travel faster than light. FTL travel can be used to travel back in time (though the procedure for doing so is somewhat involved). Sadly the apparent superluminal speed turned out to be due to experimental errors: a fibre optic cable attached improperly, which caused the apparently faster-than-...

34

Suppose you are floating in a river, and you have with you a model boat, called the SS Lightray, that can do 3 m/sec through the water. When you set the boat travelling upstream as far as you're concerned it is doing 3 m/sec. But I'm standing on the bank watching the river flowing at 1 m/sec, so when I look at your boat I see it travelling at a net speed of ...

31

In the case of relativity, "information" refers to a signal that enforces causality. That is, if event A causes event B, then some signal must travel from A to B. Otherwise, how would B "know" that A had occurred? Some examples: Light (signal) from a candle (A) hits your eye (B), causing you to see it. Electricity (signal) flows from a connected switch (A) ...

31

Spacelike separation means that there exists a reference frame where the two events occur simultaneously, but in different places. Timelike separation means that there exists a reference frame where the two events occur at the same place, but at different times. Lightlike means that, well, light could travel between those points.

30

This is an idea known as the block universe. The example of a photon is a bit extreme, since photons have no rest frames, but the idea is the same: to a fast-moving observer, our future might lie in their past, suggesting that our future already exists. However, these are purely philosophical notions. In order to see what physics has to say about it, we ...

29

A tachyon is a particle with an imaginary rest mass. This however does not mean it "travels" faster than light, nor that there's any conflict between their existence and the special theory of relativity. The main idea here is that the typical intuition we have about particles -- them being billiard ball-like objects -- utterly fails in the quantum world. It ...

28

The paths of the Feynman path integral are not actually taken. The phrase "takes every possible path" is a mangled statement of the mathematical instruction to take the integral of $\exp(-\mathrm{i}S)$ over all possible paths for the action $S$ to get the probability amplitude of something happening. It is a fact of quantum mechanics that this integral ...

26

It's a bit more complicated than that. Given any two events, there is a quantity, called the interval (also 'spacetime interval' or 'invariant interval'), denoted $\Delta s^2$, and which equals $\Delta s^2=c^2\Delta t^2-\Delta \mathbf r^2$, which determines how the two events can relate to each other causally. If $\Delta s^2>0$, then we say $A$ and $B$ ...

22

From Hacker News https://news.ycombinator.com/item?id=6253263 This is a far more interesting question than it might seem at first glance, and it deserves some attention because it tells us something fundamental and wonderful and just bloody awesome about the universe. But I don't know how to tell the story succinctly. So I'm going to do that thing I do. I ...

22

What follows is certainly not a comprehensive answer addressing all of your concerns. It is an answer to the question is there a way to see something clearly pathological like superluminal signals in the heat equation? I would argue that yes, there is. The general solution to the initial value problem $T(x,0) = T_0(x)$ for the heat equation on the real ...

22

This is really a subtle point. You are right that in 25% of the cases, Bob will randomly chose the "correct" measurement basis and thus get the correct value. However, there is no way for Bob to know when he has actually chosen the right basis and when he has chosen the wrong basis, so his measurement outcome does not contain more information that a random ...

21

In special relativity, you think of a 4-dimensional space-time. The key point here is that two events, 1, and 2 happening at $t_{1}, x_{1}, y_{1}, z_{1}$ and $t_{2},x_{2},y_{2},z_{2}$ have a distance given by ${}^{1}$ $$(\Delta s)^{2} = -c^{2}(\Delta t)^{2} + (\Delta x)^{2} + (\Delta y)^{2} + (\Delta z)^{2}$$ Now, we can therefore give any two events a ...

21

In this answer we'll basically repeat Leandro M.'s good answer for a tachyonic field using formulas. (In contrast, note that the current version of the Wikipedia page mostly discusses the hypothetical notion of a tachyonic point particle, which by the very definition moves faster than the speed of light, which is widely believed to be irrelevant for modern ...

21

It might help to cite your source: I found this one here - is this what you speak of? Anyhow, actually this kind of idea has had considerable, if not mainstream attention over the years. Many people who have worked with quantum mechanics will have at least heard of the following: it's just that it doesn't make it into many QM courses (being an equivalent ...

21

Firstly, "an equation that describes a definitive outcome as the result of some input factors" is a description of a deterministic relationship between input and output, not needfully a causal one. Causality and determinism are not the same. What if the equation is invertible, so that one can swap the roles of input and output and describe the new equation ...

21

You're right as far as it goes -- if you can come up with a Newtonian system that reaches a stationary state from a non-stationary one, then the system must be non-deterministic. The point (to the extent there is a point here) is that this is not as easy as you seem to assume it is. The vast majority of nice smooth Newtonian systems cannot reach any ...

20

Let's put it very simple: They tell you "how far something is apart compared to c" time-like: if you are fast enough, you can be at (think spatial, like "at the festival") event a and at event b, it is only a "matter of time" until you see the second event space-like: the two events are too far apart (in space). You cannot see ...

20

We can move back and forth in space, so why does the negative sign mean we can't move back and forth in time? As illustrated in the answer by Ben Crowell and acknowledged in other answers, that relative sign doesn't by itself determine which is future and which is past. But as the answer by Dale explains, it does mean that we can't "move back and forth in ...

19

Edit regarding 3+1 spacetimes and causality I'll keep adding to the answer as I get more information, and hopefully everything will just evolve along. At the very least, I'll have a set of notes to work from in the future :) This is also the first, broadest, cut at an actual answer regarding causality. Alcubierre sets out to find his warp drive metric ...

19

Physics is the discipline that studies natural phenomena, and finds mathematical models that fit the measurements and observations and also predict future behavior of the system under study. Mathematics is a discipline which studies numbers with sophisiticated methods, it has axioms and theorems and can prove statements or disprove them absolutely. To use ...

17

The force does not change instantaneously, the correct way the electromagnetic field of (and thus the force exerted by) a moving electric charge is given by the Liénard-Wiechert potential, where one can see that the effect of the charge does not travel faster than light.

17

To add to ACuriousMind's answer on the Liénard-Weichert potentials, you can put these formulas into an even more wonderfully descriptive form since you can derive Feynman's formula from them for the radiation from a moving charge: \vec{E} = -\frac{q}{4\,\pi\,\epsilon_0}\left(\frac{\vec{R}}{R^3}+\frac{R}{c}\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\vec{R}}{...

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