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We know, according relativity theory, no information or matter can travel with velocity bigger than light. However, consider the following situation: a completely empty water hose is hypothetically connected between the sun and the earth. After this, we turn on the water hose until it stay completely full, next we turn off. When we turn on again here on the earth, the water will arrive instantly on the sun in order a person there, to receive instantly the information that the water hose was turned on here on the earth. This case, the information travel with velocity bigger than light, violating the relativity theory. How can this happen?

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    $\begingroup$ The water won't come out instantly. When you turn on the hose there will be a pressure wave which will travel through the water at the speed of sound and when that reaches the sun the water will come out. $\endgroup$ – By Symmetry Feb 14 '16 at 15:34
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First of all, you have completely ignored many properties of fluids such as compressibility, viscosity, etc. One can easily argue in the macro domain saying that there would be compression of water, there would be a pressure-density wave (technically, this disturbance is a sound wave propagating through the medium whose speed if of course limited to a very small quantity) which carry the energy and what not.

If you go a little deeper, there are forces which are responsible for pushing the water molecules. These forces are carried by particles called force carriers which are just packets of energy. But whatever the particle maybe, it must obey the laws of physics which means, it cannot go any faster than the speed of light. Hence, there would be a time lag before one would see the effect at the sun.

You can also argue that, the liquid is made up of atoms and all of the atoms in the liquid cannot all of a sudden start moving together. Every atom must tell the adjacent atom that "Hey! I am getting closer to you, We don't like each other! Move away" and hence something similar to a chain reaction takes place. The atom basically exerts an electromagnetic force to repel the atoms which get too close to it. This electromagnetic force is carried by photons, which is light and travels at the speed of light.

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Water is compressible (and stretchable). Disturbance, in your case, some form of sound, travels at most as fast as sound in water. This is about million times slower than light. The same with steel rod.

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