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The weak equivalence principle is (from the Weinberg book)

At every space-time point in an arbitrary gravitational field it is possible to chose a 'locally inertial coordinate system' such that, within a sufficiently small region of the point in question the laws of motion of freely falling particles take the same form as in an unaccelerated coordinate system in the absence of gravitation.

Weinberg than says later on that the weak equivalence principle is just the restatement of the equivalence of inertial and gravitational mass. I don't understand this. He also makes an example. He considers an object that falls towards the Earth. And then he considers a coordinate system that moves with $x'=x-gt^2$. In this coordinate system the object that moves towards Earth is at rest. Weinberg says that this is because there is a cancellation between gravitational and inertial forces!? I cannot see this explicitly...

The gravitational force between two gravitational masses $m$ and $M$ is $F_g=G\,m_g\,M_g/r^2$.

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The equivalence of inertial mass $m_I$ and gravitational mass $m_G$ implies that the universality of free fall, and this universality is identical to inertial force. So we could view gravitational force as inertial force as well. As a consequence, as Einstein reflected, we should not be able to determine the existence of gravity in a freely falling elevator in a homogeneous static gravitational field as we have eliminated it in such a frame. In such a frame, all freely falling objects are traveling in a straight line. According to this view, it turns out that the inertial frames should be those who are freely falling in the gravitational field. However, the difference between gravitational force and inertial force is that the former has inhomogeneity (and is also time changing too) whereas the latter does not. Because of that, we can only eliminate the gravitational force in a freely falling elevator in a sufficiently small region (locally inertial coordinate system). Hence, the weak equivalence principle is equivalent to the equivalence of $m_I$ and $m_G$. When an observer is freely falling, whose coordinate system is $x'$ as in your question, there is a gravitational force pushing him towards the source while at the same time, because he is accelerating, there is an inertial force that is opposite direction with the gravitational force, so the cancellation of both forces and make him an inertial frame.

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  • In GR even an accelerated frame of reference (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. The fact they can't be distinguished comes from inertial and gravitational mass being the same. This elucidated in the conclusion after case 2. Rest of cases are for the sake of completion.

  • 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.This possible because the inertial and gravitational masses are same!! Hence, the acceleration they experience by gravity can be replaced by pulling.

  • 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 ( International space station) . 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.

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