Why do balls follow curved paths in Earth's gravity according to GR?

[Antenna Guy]

We can also add a third inertial observer C and note the following:

1) A and B agree at all times on matters of simultaneity.
1) If A and B agree that two events are simultaneous then C does not agree those events are simultaneous.

bzzzt - not necessarily.

Let A and B be displaced equidistant from an origin O along x.
Let C (and D) be the same distance from O along y.
Let two events that appear as simultaneous to A and B occur on the z-axis.
C (and D) will not only agree on simultaneity, they will also agree with A and B regarding the time that the simultaneous events occurred (relative to O).

Regards,

Bill

 

[yuiop]

bzzzt - not necessarily.

Let A and B be displaced equidistant from an origin O along x.
Let C (and D) be the same distance from O along y.
Let two events that appear as simultaneous to A and B occur on the z-axis.
C (and D) will not only agree on simultaneity, they will also agree with A and B regarding the time that the simultaneous events occurred (relative to O).

Regards,

Bill

In the particular situation we were discussing, A and B are non inertial observers onboard an accelerating rocket and are spatially separated along the axis parallel to the accelerating motion of the rocket. Both A and B feel and measure proper acceleration. C is an inertial observer not on board the accelerating rocket. If you wish to introduce a fourth inertial observer D that is spatially separated from C, then C and D should be on a line parallel to the line joining A and B. The situation we were discussing is the Rindler spacetime drawn on a Minkowski diagram with one space dimension and one time dimension. Under those conditions, my original assertion:

1) A and B agree at all times on matters of simultaneity.
1) If A and B agree that two events are simultaneous then C does not agree those events are simultaneous.

remains true and can be extended to :

1) A and B agree at all times on matters of simultaneity.
1) If A and B agree that two events are simultaneous then C and D do not agree those events are simultaneous.

Your counter argument is based on all the observers (A,B, C and D) being inertial observers which is not the situation that was under discussion.

 

[yuiop]

Hi again kev. I understood what Mallinckrodt wrote as: the rod flashes at the same clock times all along the rod, but for rear- and front end observers the intervals between the flashes are different. It should be like that if the proper length remains the same for those observers, yet they experience different proper accelerations. The front clock records less acceleration because it runs faster than the rear clock. How else?

-J

Hi Jorrie,

I am at a slight disadvantage because for some unusual reason my browser does not display any of the diagrams or equations in the Powerpoint presentation you linked to :(

The exact quote from Mallinckrodt was "Note that within the frame of the rod, flashing synchronously is not the same as flashing at a definite time interval because the clocks run at different rates."

His statement is a little vague without being able to see the diagram. If I add comments in brackets his statement reads as "Note that within the frame of the rod, flashing synchronously is not the same as flashing at a definite time interval (in which frame, accelerated or inertial?) because the clocks run at different rates.(in which frame, accelerated or inertial?"

The first part is clear. The flashes occur simultaneously within the frame of the rod.

The second part is not clear. The clocks run at different rates is an observation made by an inertial observer not within the frame of the rod. As far as I am concerned, the clocks run at the same rate as measured by accelerating observers within the frame of the accelerating rod.

For example if observer A is onboard the rocket (near the tail) and sends signals at intervals of one second as measured by his local clock, then an observer B also onboard the accelerating rocket but near the nose will detect the signals as arriving at intervals of once per second using his local clock. The same happens if B sends signals to A in the opposite direction. By Einstein's definition the clocks are synchronised as far as observers A and B are concerned.

Now what is a little puzzling is that the signals will appear to be redshifted to A and blue shifted to B. The prof seems to be wrongly assuming that a redshifted signal automatically implies the clocks are running at different rates. This is not always true. What is true is that velocity of the emmitter at the time the signal was emmited was slower than the velocity of the receiver at the time the signal was received. as viewed by the inertial observer outside the rocket. Remember the observers inside the rocket can consider themselves to be stationary within a gravitational field so they do not consider themselves to have a velocity.

As mentioned before, observers A and B can prove to themselves that their clocks run at the same rate and remain synchronized. Knowing that, presumably they must conclude that the wavelength of the light wave is length contracted as it falls from the nose to the tail in the apparent "gravitational field" they are experiencing.

 

[Jorrie]

His statement is a little vague without being able to see the diagram. If I add comments in brackets his statement reads as "Note that within the frame of the rod, flashing synchronously is not the same as flashing at a definite time interval (in which frame, accelerated or inertial?) because the clocks run at different rates.(in which frame, accelerated or inertial?"

Hi kev, I attach .JPGs of the ST diagrams he referred to; hope they're viewable. I think he meant the "instantaneous comoving inertial frame", in which the two accelerating observers are at rest for a moment (x-axis along the line of "instantaneous simultaneity" through the origin).

Remember the observers inside the rocket can consider themselves to be stationary within a gravitational field so they do not consider themselves to have a velocity.

But, in their 'apparent uniform gravitational field', their clocks would run at different rates, not so. I think we have a problem with what 'simultaneity' means along a lengthwise accelerated rocket.

Prof. Mallinckrodt also said in his conclusion:

• Observers in a rigidly accelerated frame agree on almost everything except what time it is. Clock rates are proportional to vertex distances.
• When a rigid body returns to the velocity at which all of its clocks were synchronized, the clocks regain synchronization.
• I believe this material is accessible, surprising, uncontroversial, but nevertheless not well known.

I still have a feeling that the prof is right...

 

[yuiop]

Hi kev, I attach .JPGs of the ST diagrams he referred to; hope they're viewable. I think he meant the "instantaneous comoving inertial frame", in which the two accelerating observers are at rest for a moment (x-axis along the line of "instantaneous simultaneity" through the origin).

The observer in the "instantaneous comoving inertial frame" does not make the same measurements/ observations that the accelerating observers make. The first diagram you posted shows that the rocket stops accelerating and as you know, I have already concluded in another thread that when the rocket stops accelerating the clock will no longer be syncronised in the rocket frame. While the rocket continues to accelerate indefinitely, the accelerating observers onboard the rocket will consider the clocks to remain synchronized.

But, in their 'apparent uniform gravitational field', their clocks would run at different rates, not so.

I agree that in a curved gravitational filed such as on the Earth that a clock higher up a tower would appear to run faster than a clock lower down. It would be impossible to get two such clocks to remain sychronised without artificially processing the rate of one of the clocks to slow it down or speed it up. In the rocket the same is not true. The 'apparent uniform gravitational field' they observe on the rigidly accelerating rocket is not the same as the curved gravitational field of the Earth. Another way of expressing the spacetime onboard the rocket is saying it is equivalent to a parallel uniform gravitational field to separate it from a curved gravitational field, which is less confusing than calling it a flat spacetime that seems to suggest to some people that no acceleration or gravity is present.

I think we have a problem with what 'simultaneity' means along a lengthwise accelerated rocket.
...

Well, I am using Einstein's definition of simultaneity where clocks are syncronized by sending timing light signals, as I posted earlier. I am also interpretating the "line of simultaneity" within the rocket frame to mean the line along which observers in the accelerated frame consider events to be simultaneous. If signals are sent simultaneously (as measured by observers in the rocket) then they arrive simultaneously (as measured by the observers onboard the rocket), because the signals arrive on the same on the same line of simultaneity.

I have a feeling this question might have to become a thread of its own :P

 

[Antenna Guy]

In the particular situation we were discussing, A and B are non inertial observers onboard an accelerating rocket and are spatially separated along the axis parallel to the accelerating motion of the rocket. Both A and B feel and measure proper acceleration.

O.K.

C is an inertial observer not on board the accelerating rocket.

Did I say that?

If you wish to introduce a fourth inertial observer D that is spatially separated from C, then C and D should be on a line parallel to the line joining A and B.

I'm fairly certain that C and D need only accelerate with, and in the same direction as A and B. However, since I did not accurately represent what you were calling C, would it be O.K. to let C and D be co-located with events that A and B agree are simultaneous?

The situation we were discussing is the Rindler spacetime drawn on a Minkowski diagram with one space dimension and one time dimension.

O.K.

Under those conditions, my original assertion:

remains true and can be extended to :

1) A and B agree at all times on matters of simultaneity.

Consider this: with only one space dimension to work with, any pair of simultaneous events would necessarily have to occur at the same point. Overlooking this, A and B will certainly agree upon the simultaneity of any two such events, but they will not observe the events simultaneously.

1) If A and B agree that two events are simultaneous then C and D do not agree those events are simultaneous.

How so? If A and B agree that any two co-located events are simultaneous, why wouldn't/couldn't any number of additional observers agree on the same thing?

Your counter argument is based on all the observers (A,B, C and D) being inertial observers which is not the situation that was under discussion.

Hmmm.. If I modify what I stated and let C and D simply have a relative displacement that is perpendicular to, and symmetric about the instantaneous velocity of A and B, I think you will find that all four can agree on simultaneity (although they may disagree on time).

Regards,

Bill

 

[yuiop]

"C is an inertial observer not on board the accelerating rocket. "

Did I say that?

No I said that. I thought I made it clear when I said:

We can also add a third inertial observer C and note the following:

1) A and B agree at all times on matters of simultaneity.
1) If A and B agree that two events are simultaneous then C does not agree those events are simultaneous.

I'm fairly certain that C and D need only accelerate with, and in the same direction as A and B. However, since I did not accurately represent what you were calling C, would it be O.K. to let C and D be co-located with events that A and B agree are simultaneous?

I specified that C is inertial and therefore not accelerating.

Consider this: with only one space dimension to work with, any pair of simultaneous events would necessarily have to occur at the same point.

We can have a one dimensional line and we can have two events that are not at the same point on that line, that can be considered as simultaneous by observers at rest with line. An observer not at rest with the line will not think the two spatially spearated events are simultaneous.

How so? If A and B agree that any two co-located events are simultaneous, why wouldn't/couldn't any number of additional observers agree on the same thing?

We were not talking about co-located events.

Hmmm.. If I modify what I stated and let C and D simply have a relative displacement that is perpendicular to, and symmetric about the instantaneous velocity of A and B, I think you will find that all four can agree on simultaneity (although they may disagree on time).

Regards,

Bill

Yes, if C and D are accelerating along with A and B, but I specified C and D are inertial observers and they are not accelerating.

 

[Jorrie]

I have a feeling this question might have to become a thread of its own :P

I agree; this thread is becoming too long. Will you start one and state the question around a ST diagram?

 

[Jorrie]

Well, I am using Einstein's definition of simultaneity where clocks are syncronized by sending timing light signals, as I posted earlier.

Hi kev, just a two points: AFAIK, Einstein's method for synchronizing clocks only works in inertial frames, where the speed of light is isotropic, which is not the case in an accelerated frame.

Secondly, I think "agreeing on simultaneity" in the accelerating frame does not necessarily mean that their clocks agree on what time it was when an event happened.

-J

 

[yuiop]

I agree; this thread is becoming too long. Will you start one and state the question around a ST diagram?

No need. I have figured out my mistake and it turns out you (and the prof) are right. My mistake was my incorrect assumption that the line of simultaneity joined points of equal proper time in the rocket frame. having done some detailed calculations, the clock higher up the rocket not only run faster according to the inertial observer outside the rocket but also from the point of view of the accelerated observers inside the rocket. The "line of simultaneity" only joins points of equal velocity in the time space diagram of the accelerating rocket. It is not the same as the line of simultaneity in the inertial case with constant relative motion. As you also mentioned the speed of light is not isotropic to the accelerated observers. Dang!