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The general theory of relativity describes the gravity as a geometrical feature (in very twisted geometry). That's comprehensible to me when the motion occurs and "the most straight" trajectory is to be found. I have a problem comprehending the interactions which would classical dynamics describe as static (such as water pressure due to the gravity). I think the path leads through the existence of $ct$ dimension, but I am not able to finish off the reasoning. Could someone help?

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There are various ways to approach this, but my favourite approach is to say that everything is ultimately made up of particles, and composite bodies like solids and fluids are just assemblages of particles.

The reason why this makes understanding pressure simpler is because for a single point particle the stress-energy tensor has a very simple form. It is:

$$ T^{\alpha\beta}({\bf x},t) = \gamma m v^\alpha v^\beta $$

at the position of the particle and zero everywhere else. The variable $v$ is the velocity vector $(c, \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt})$ i.e. it is the derivative of the position with respect to coordinate time (not proper time).

To understand why the diagonal elements behave like a pressure observe that they have the form $\gamma m(v^i)^2$ and this looks like a kinetic energy. So for the fluid/solid/whatever the total $T^{ii}$ terms arise from summing the kinetic energies of gazillions of point particles all moving randomly, and this is exactly what we mean by pressure in a gas.

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John Rennie does a good job talking about pressure in general relativity which address your title question. In the body you asked about situations that in Newton would be static.

So first note that in general relativity the inertial frames are the frames of freely falling objects. So imagine a falling elevator. What does it see about the static water pressure.

It sees a layer of water see some higher pressure at the bottom of the layer and less pressure at the top of the layer. So it sees a net push on the layer of water that accelerates it upwards.

So in the frame of the falling elevator, the layer of water feels an actual net force and it accelerates upwards at $9.81m/s^2.$ Every layer of air and water and dirt feels a smaller pressure from above than from below and feels a net force up of exactly enough to accelerate it upwards at $9.81m/s^2.$

Even a person standing on the earth feels a pressure upwards on their feet (the earth underneath is compressed enough to give that pressure) that exceeds the pressure of the air above them by exactly enough to accelerate the person upwards at $9.81m/s^2.$ And that's also why you feel a pressure on your feet.

So the person on the falling elevator sees everything else accelerate upwards at $9.81m/s^2.$ And it sees that happen from actual forces from actual pressure differences. And of course all those accelerating things see each other being at rest relative to each other. And some things, such as baseballs in flight don't have this net force on them, so the guy on the elevator sees them move in a straight line since they don't have a force on them.

So the elevator sees things in free fall move in straight lines and it sees most things accelerate upwards at $9.81m/s^2.$ So most things are "at rest" when the inertial frames know they are actually accelerating upwards at $9.81m/s^2.$

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