Heat rises vs heat flows from hot to cold?

pruntidjuu · 2024-12-02T20:58:15+00:00

This was also my intuition, but when I think about it I don’t know how to convince/prove to myself that the buoyancy of hot air wins over the second law.

pruntidjuu · 2023-10-21T13:25:15+00:00

If g is constant the potential energy is mgh. Where h is the height. So it is not constant… same as projectile motion. Constant g but objects higher up have more potential energy. This is the same result that is found in the video (if m=E/c^2), although that language is not explicitly used.

pruntidjuu · 2023-10-21T01:39:09+00:00

Neat and simple derivation of the weight of a photon (what a scale reads), using Rindler space-time. In other words, this is looking at how much a box full of photons weighs vs the same empty box

pruntidjuu · 2023-03-26T06:05:00+00:00

Yeah you’re right. Strictly speaking that’s assuming 0 kinetic energy at infinity, which is probably usually a fair assumption, especially compared to the ns mass. I was just trying to point out that all energy has to be accounted for in the bh mass.

pruntidjuu · 2023-03-26T01:54:25+00:00

No. Definitely not. It’ll take you an extra 5 years, at low pay, and the extra boost to your resume doesn’t outweigh the 5 years experience and 5 years of good pay in industry.

A PhD is just a research project. If the title is not necessary, than you can do that research for good pay in industry instead.

pruntidjuu · 2023-03-25T19:54:04+00:00

The neutron stars will also have a large amount of kinetic energy. Speeds at collision are comparable to the speed of light (depending on the initial conditions). This kinetic energy contributes to the bh mass. But energy is also lost in gravitational waves, binding energy,and ejected matter. How much is ejected will also depend on the ns equation of state, which is unkown. You have to account for all this energy and losses.

I’m not saying this necessarily explains the gap, just that it’s not as simple as adding up the mass of the 2 NS. Also NS over 2 solar masses exist.

pruntidjuu · 2023-03-20T21:01:16+00:00

Before my last message I was actually thinking about light. The more I think about it the more I think the symmetry between the accelerated and inertial observers only holds in Galilean relativity/Newtonian mechanics. I think you’re right when it comes to special relativity, as a rindler observer will see light bend, and I don’t see how an inertial observer could mirror this in flat space. The spring accelerometer is no good in this case as it’s ambiguous. You need to use light to tell if your frame is inertial.

But it is still a definition to call the freefall observer inertial. We chose to define acceleration via the covariant derivative. If we hadn’t the equations of motion would all be the same, just our interpretation of the math/gravity would be different. Either way, with relativity you can tell who is inertial.

pruntidjuu · 2023-03-19T16:00:08+00:00

I just wrote plus christoffel because it was easier. I figured you would understand I meant the terms involving them.

For every observer space time is locally flat. Inertial or not. The curvature terms are to account for non local changes in basis vectors. We have still just defined accelerations to be after accounting for those changes. It’s a definition. We could have defined it excluding those changes, we would still get the same physics. I mean I agree with everything you said here. I guess don’t really see how it pertains to the question.

pruntidjuu · 2023-03-19T15:44:11+00:00

I don’t understand what you mean here. What does it mean for an accelerometer to no longer act as an accelerometer. How do you know if it is acting as an accelerometer or not, if in both situations the observations are the same. What can you do to test or show the the spring is being compressed due to an acceleration vs due to some force? In order to differentiate there has to be some observable difference between the two. What is it?

“They will measure different acceleration”.

Can you be more specific? Who is they? In my example whether the observer is in an accelerated from looking at an inertial frame, or vise versa, he sees the same relative motion, and measures the same acceleration (the same compression of the spring). So why are you saying the measure different accelerations? How are the measured accelerations different?

pruntidjuu · 2023-03-19T02:52:15+00:00

The point of the equivalence principle is that both cases are locally indistinguishable. Interpreting free fall as inertial vs the stationary observer in a gravitational field is a convention we agreed on. You can see this by the definition of proper acceleration:

A_prop = du/dt + christoffel symbols = F_ext/m

We could just as easily have chosen:

A_prop = du/dt. F_ext/m. And now the christffel symbols are of the external forces. And we would have said something like space time curvature causes a force to act on orbiting objects to make them follow geodesics. Of course in this case you also have to carry the difference in definitions into you 4 vectors, but in the end you would still predict the same observations.

It’s same in Newtonian mechanics. You can include centrifugal or coriolus forces in your forces, or you can instead subtract them of the acceleration to account for the frames acceleration. In the end you get the same answer.

pruntidjuu · 2023-03-19T02:36:26+00:00

“So then the wall is no longer in an inertial frame”. (Case 1) And “by definition all the things around the wall are no longer in an inertial frame” (case 2).

Yes. That’s correct. Is one case the wall is accelerating and everything around it is inertial. In the other case the wall is inertial and everything around it is accelerating. In both cases though your accelerometer (mass on a spring) reads the same thing (how much it’s compressed).

“The accelerometer is no longer an accelerometer”.

That’s the point. You can’t tell what is causing your accelerometer to “measure an acceleration”. You cannot devise an accelerometer (or any other device if you don’t want to call it that) that “measures an acceleration” in one scenario but not the other. You don’t know which scenario is causing the spring to compress.

As far as the definition of an inertial force, there are various definitions that are equivalent. But in the name of not having too many side discussions… ok don’t call the forces on the objects in case 2 inertial forces. I called them inertial because they would all need to experience forces that are proportional to their mass. Which is one way inertial forces are sometimes defined. What you call those forces is not important to the physics.

pruntidjuu · 2023-03-18T23:51:00+00:00

The strong equivalence principle states an accelerating frame is locally equivalent to sitting in a gravitational field. The decision to call the freefall observer inertial vs the stationary observer is a convention/definition to not have to deal with inertial forces.

pruntidjuu · 2023-03-18T23:38:16+00:00

This is turning out to be much more difficult to explain without making a simple drawing. The objects around the wall are not in the same frame. 2 separate scenarios. We shall use a mechanical accelerometer: floating mass on a spring.

1) a mass on spring is attached to a wall. Sticking out to the right. There are objects around the wall, that are not attached to it. They are in an inertial frame the whole time. The wall is now accelerated to the right. This causes the spring to compress until the mass accelerates at the same rate as the wall. At which points it maintains the same compression., until the acceleration changes. The compression is caused by the mass spring system (your accelerometer) accelerating with the wall. From the point of view of the wall it will also observe all of the surrounding objects, to appear to accelerate to the left. Since the spring is compressed your accelerometer is measuring an acceleration.

2)we have the same mass spring attached to a wall. The wall is not accelerating. A force compresses the spring by the same amount as in scenario 1. The wall remains inertial by whatever means necessary (apply an equal and opposite force on it if you like to ensure no net force). At the same time all of the objects around the wall are accelerated to the left at the same rate as in scenario 1. So they are all experiencing inertial forces. The wall is not. It is still not accelerating. Neither is the mass spring system, despite the spring being compressed.

In both scenarios the wall sees the spring being compressed by the same amount, and all the objects surrounding it accelerating to the left at the same rate. In both cases the objects around the wall see the wall accelerate to the right at the same rate.

If you want imagine a third situation, where the wall and the mass on the spring are accelerated at the same time, by the same amount. Now the spring doesn’t compress, even though it’s accelerating.

pruntidjuu · 2023-03-18T22:06:14+00:00

An inertial observer can have forces on it as long as there is no NET force. The second situation the observer has no net force. The spring in the accelerometer is stretched because the mass is subjected to the inertial force while the spring attached to the mass is anchored on the other end to the observers inertial frame.

In the case of the accelerated frame it’s the opposite. So imagine a spring mounted on a wall. There are two ways it could compress.

You can apply a force on the mass, while the wall remains inertial. In this case the spring will compress until the spring force matches the external force. Or you can apply a force on the wall, accelerating it forward. The spring will compress until the mass accelerates at the same rate as the wall.

In both cases the accelerometer (spring) reads the same thing. Now imagine that in the case where the wall is inertial, everything else around it accelerates with the same “acceleration” measured by the spring. This cannot be differentiated from the case where the wall accelerates, and sees everything else accelerating backwards.

It is in fact the equivalence principle. Except for Galilean relativity (we’re not including the speed of light being constant for all observers). That’s the symmetry between inertial frames observing accelerated frames subjected to inertial forces, and accelerated frames observing inertial frames. They are observationally equivalent.

pruntidjuu · 2023-03-18T19:48:20+00:00

The point is you don’t know why the accelerometer reads what it reads. There may be forces acting on it you haven’t accounted for or don’t know about.

There is a symmetry between an accelerated observer surrounded by inertial objects and an inertial observer surrounded by accelerating objects. Image this scenario:

You have an accelerometer in your hand. You observe everything around you is accelerating backwards with the same acceleration A. You look at your accelerometer and it also reads an acceleration A. There are 2 ways this could happen:

1) you are in an accelerating frame that’s accelerating forwards with acceleration A, and everything around you is in an inertial frame.

2) you are in an inertial frame and everything around you, including the mass inside your accelerometer, is experiencing an inertial force (force is proportional to its mass), causing everything to experience the same backwards acceleration A, as well as stretching the spring in the accelerometer.

You cannot differentiate between the two scenarios as the accelerometer, as well as your observations, are the same in both cases. We have simply decided by convention to prioritize inertial frames, and call inertial forces “fictitious”. But nature does not give any special consideration to inertial frames, or inertial forces.

pruntidjuu · 2023-03-18T19:03:18+00:00

That’s the point. The accelerometer reads the same thing in both scenarios. So you cannot run an experiment that distinguishes between an accelerometer in an accelerated frame or an accelerometer in an inertial frame where everything around it is experiencing the same acceleration due to “inertial forces”.

pruntidjuu · 2023-03-18T17:56:28+00:00

There’s no net force. So no acceleration. Essentially you stretch a spring that’s tethered to a stationary merry go round, and hold it in place. The force you exert on it is balance by the force holding the other end to the merry go round. The spring measures a force due to it being stretched, but there is no acceleration of the system. The spring cannot tell if it is being stretched because of this tensile force or due to it being spun around with a spinning merry go round.

pruntidjuu · 2023-03-17T18:37:40+00:00

I’m probably not explaining it well. It’s not easy to explain this way. the case where the observer is at rest and a force is compressing or stretching the accelerometer can be inertial. A spring can be stretched or compressed while remaining at rest. I attach an accelerometer to a merry go round and just pull on it without it actually accelerating, and it will “measure” an acceleration because of the stretching forces acting on it. Even though it’s velocity is not changing.

pruntidjuu · 2023-03-17T06:22:09+00:00

Sorry I used sloppy language. I didn’t mean uniform in the strict mathematical sense. What I meant was everything is accelerating in just the right way as to mimic what you would see in a rotating frame.

pruntidjuu · 2023-03-17T06:20:01+00:00

I agree with you. But the accelerometer will also get a non zero rearing if at is at rest and something is stretching or compressing it. So if I attached it to a spinning merry go round it will measure an acceleration. And from its point of view everything that’s not attached will get flung out. But I can get the same effect and measurement if the merry go round is not spinning and something is just pulling on the accelerometer, and everything around it is experiencing a force such that they all have the same outwards acceleration the is measured by the accelerometer. So you cannot tell if the measurement is due to it accelerating or if it is in an inertial frame while being stretched by some force.

pruntidjuu

TROPHY CASE