Logic of general relativity

Question

I’m a beginner in general relativity, I start with reading Field Theory written by Landau. There are some confusions arose when I tried to understand the physic logic of general relativity. Let me first conclude how general relativity is built up in this book:

1. Truth: In inertial reference frame, we know a truth of gravitation field, that is, no matter what mass does matter have, all objects will follow exactly the same trajectory under the action done by gravitation field if they start with the same initial conditions.
2. Equivalence principle: the above phenomenon also happens for free objects when observed by a non-inertial frame, thus, we can view non-inertial reference frame as a inertial reference frame equipped with a gravitation field (of course, this field may not be real field generated by real matter).
3. Geometry of non-inertial frame: we know that in general, the space-time distant in non-inertial frame is described by $ds^2=g_{ij}dx^idx^j$, where $g_{ij}$ is metric different from Minkowski metric.
4. Geometry fact of gravitation field: use the equivalence principle, we then conclude that the changes caused by gravitation field in inertial frame is exactly the distortion of 4-dim spacetime geometry, and $g_{ij}$ will be the measure of gravitation field.

My confusions then follows:

When we equipped a gravitation field with an inertial frame, the above tells us it will no longer be inertial frame, since the space-time distance is no longer described by Minkowski metric. Now, this logic tells me, there does not exist inertial frame that you can observe gravitation field, which is a contradiction of the truth 1, where we start from a inertial frame and state a truth about gravitation field. What’s wrong here?

Answer 1

I think that the situation is easier to understand for an uniformly accelerated frame. For an observer in this frame the metric is not the diagonal (1 , -1 , -1 , -1). But if he "jumps" to be in free fall, he can be momentarily comoving with the frame, being in a Minkowsky spacetime, where the path of the accelerated frame can be calculated without limitation of time or space.

The only difference in the case of a gravitational field is that the "jump", while also let him momentarily comoves with the frame, results only in a local Minkowski spacetime.

For example, the ISS frame will calculate everything moving in straight lines and constant velocities in a very small $\Delta t$. Beyond that, the earth rotates around it for example, what is not compatible with a Minkowski spacetime. Any calculation using his free fall coordinates must use the relevant spacetime metric to match the observed events.

Answer 2

I am assuming your confusion stems from your understanding of what is an inertial frame of reference (FOR) in GR.

Inertial FOR in different fields in physics:

Classical Mechanics (CM):

• An inertial frame of reference (FOR) in classical mechanics is one that moves with a constant velocity. Essentially, an accelerated FOR is non-inertial in CM. Take an example of a pendulum hanging inside a train moving with a constant velocity. Here, the pendulum is stationary unless acted upon by a force from something inside the train--obeying the Newton's 1st law. However, when the train is suddenly uniformly accelerated we see that the pendulum moves, and hangs at angle. This can not be explained by any force within a FOR inside the train. As for an observer inside the train suddenly sees the pendulum move without any force acted upon it. Hence, it breaks the Newton's 1st law! Therefore, an accelerated FOR is non-inertial in CM.

General Relativity (GR):

• In GR even an accelerated FOR is inertial. This is because one can choose a free-falling FOR in GR, and the physics still remains intact. This is possible because locally one can not distinguish between the acceleration and gravity.

• Case 1:Consider somebody in a small box floating in the outer space carrying two stones, this person and the stones are free floating. Now, if somebody outside starts to pull this box with a constant acceleration the person, and stones are pressed to the floor.

• Case 2: Similarly, if the box is brought close to a constant gravitational field. Again the person and stones are pressed to the floor of the box.

• There is no experiment that a person can do inside the box to distinguish which was due to gravity, and which was a result of pulling upwards on the box.

• Case 3: Furthermore, if the box is free-falling under a constant gravitational field. The person and stones will still be floating like in the empty space.

• There is no experiment that a person can do inside the box to distinguish which was due to floating in empty space, and which was a result of falling under a gravitational field. Hence, one can switch off gravity by moving into free falling FOR.

• Case 4: Now, if the box is brought close to the Earth then both the person and stones are pressed to the floor. In this case, since the gravity is radially pointed towards the center of Earth unlike case 3. There is a possibility that the experimenter inside the box can drop both the stones with separated distance at the same time. Assuming the box is big enough one can see that the stones comes close to each other as they fall. This implies one can distinguish between gravity and force if the FOR is large enough!

• To conclude, the gravity can be switched off in a locally free-falling FOR. Gravity distorts spacetime which is measured by the general metric g_ij. In addition, one can choose a local enough point on this metric that is free-falling and does not experience any gravity, which is quantified by the Minkowski's metric. An example of this is the ISS (as pointed out in the above answer). In a FOR that is large enough to include Earth and ISS one can see the spacetime distortion by gravity, and is quantified by g_ij. However, one can also choose a local enough FOR on this metric such as an FOR inside the ISS, where gravity experienced is zero, and mere minkowski metric is sufficient explain the physics inside the ISS. Objects thrown inside ISS move in straight lines as expected from Minkowski's metric but to an observer on Earth they are orbiting the Earth as expected from g_ij.