# Hartle, Gravity pg 34: Question about Time in Non-Inertial Frames

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• Kashmir
In summary, proper time is the time measured by a clock and coordinate time is the time coordinate used in coordinate systems.
Kashmir
Hartle, gravity
pg 34

" An observer in a inertial frame can construct a clock that measures the time t
"*)Is time something else that exists irrespective of clocks? What we do is just that we measure it? Pg 35
" ...It's a central assumption of Newtonian mechanics that there is a single notion of time for all the inertial observers"

What does the above sentence exactly mean?
Does it mean that whatever time is, it's same for all inertial frames? So if I take similarly made contraptions that serve as clocks, to every inertial frame,they would all tick at same rate with respect to one another?

What about non inertial frames? Why were non inertial frames excluded from
"...that there is a single notion of time for all the inertial observers".

Clocks directly measure proper time, usually denoted ##\tau##. If you know the relationship between proper time ##\tau## and some coordinate time ##t## then you can use the measurement of ##\tau## to calculate ##t##.

Kashmir said:
Is time something else that exists irrespective of clocks? What we do is just that we measure it?
That is a bit philosophical. Without diving into the metaphysical connotations of "exists", proper time is the measurand of a clock. In other words, it is the thing that is measured. This contrasts with the measurement of a clock, the actual value obtained by the clock.

Kashmir said:
Pg 35
" ...It's a central assumption of Newtonian mechanics that there is a single notion of time for all the inertial observers"

What does the above sentence exactly mean?
It means that Newtonian mechanics is wrong.

Demystifier, PeroK, topsquark and 1 other person
Kashmir said:
What about non inertial frames? Why were non inertial frames excluded from
"...that there is a single notion of time for all the inertial observers".
Non inertial frames were not excluded. It is simply a statement about inertial frames - which are the frames in which Newton's laws hold good.

vanhees71 and topsquark
Dale said:
Clocks directly measure proper time, usually denoted ##\tau##. If you know the relationship between proper time ##\tau## and some coordinate time ##t## then you can use the measurement of ##\tau## to calculate ##t##.That is a bit philosophical. Without diving into the metaphysical connotations of "exists", proper time is the measurand of a clock. In other words, it is the thing that is measured. This contrasts with the measurement of a clock, the actual value obtained by the clock.It means that Newtonian mechanics is wrong.
what is proper and coordinate time? Can you please explain it.
Thank you :)

PeroK said:
Non inertial frames were not excluded. It is simply a statement about inertial frames - which are the frames in which Newton's laws hold good.
A pendulum in an accelerating frame has different time period, so doesn't the clock move differently in this frame than in an non inertial one? So is time different for the non inertial one??

Kashmir said:
what is proper and coordinate time? Can you please explain it.
Thank you :)
Get a textbook on special relativity!

The Bill, Vanadium 50 and Kashmir
Kashmir said:
A pendulum in an accelerating frame has different time period, so doesn't the clock move differently in this frame than in an non inertial one? So is time different for the non inertial one??
No. Not in Newtonian physics.

PeroK said:
No. Not in Newtonian physics.
If a pendulum is hung in an accelerating elevator and won't it's period change?

Kashmir said:
If a pendulum is hung in an accelerating elevator and won't it's period change?
For this reason, a pendulum clock is not an accurate time keeping device. Calling it a "clock" and treating it as a perfect reference for elapsed proper time in a circumstance where it is subject to variable proper acceleration is erroneous.

We do not expect a Grandfather Clock placed on the moon to tick at its nominal rate.

Put it on the surface of the Earth in reasonably constant gravity and calibrate it appropriately for the location where it is placed and you have yourself a clock.

Wikipedia has a paragraph on the effect of acceleration on clocks: https://en.wikipedia.org/wiki/Time_dilation#Clock_hypothesis

vanhees71, PeroK, Vanadium 50 and 1 other person
Kashmir said:
what is proper and coordinate time? Can you please explain it.
Thank you :)
Proper time is the time as measured by a correctly functioning physical clock. It includes the usual clocks and also other time-dependent physical processes like biological age or radioactive decay. It is defined only along the worldline of the clock and does not involve any notion of simultaneity. Each clock has its own proper time.

Coordinate time is the time coordinate in an arbitrary coordinate system. This is the time that defines simultaneity. It is the time involved in coordinate transforms, like the Lorentz transform. Coordinate time is not required to have any specific relationship to any physical clock, but even when (by some convention) it does match a physical clock’s time the coordinate time is different from the proper time: the proper time is only defined on the worldline of the clock and the coordinate time is defined throughout the coordinate chart.

Last edited:
cianfa72, vanhees71 and topsquark
Kashmir said:
what is proper and coordinate time? Can you please explain it.
Thank you :)
In addition to the earlier explanations, this might help:

On a position-vs-time graph,
the coordinate time is the "time" along the time-coordinate axis of the reference frame drawing the graph
whereas
the proper time [of an astronaut's worldline] is like the "arc-length of the curve of along the worldline"
(where "arc-length along the curve" on this diagram is measured by the astronaut's wristwatch).

In the position y-vs-time t graph below (t running to the right),
the coordinate-time of this graph is measured along the horizontal red line (y=0)
[which can somehow be "spread over space" to assign a coordinate-time to any point-event on the graph].

For these equal coordinate-time curve segments (from coordinate-time 0 to coordinate-time 2):
• in Galilean physics (PHY 101), the three worldlines have equal elapsed proper-times.
• in special relativity,
the red worldline segment has the longest elapsed proper-time.
The green worldline segment has a shorter elapsed proper-time.
The blue worldline segment has the shortest elapsed proper-time among the three.
• in Euclidean geometry, with the plot below interpreted as a y-vs-x graph (x running to the right),
arc-lengths are measured by each path-surveyor's odometer.
The red curve segment has the shortest arc-length.
The green curve segment has a longer arc-length.
The blue curve segment has the longest arc-length among the three.

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Dale and topsquark
Kashmir said:
If a pendulum is hung in an accelerating elevator and won't it's period change?
Yes, but a quartz wristwatch will still keep the same time.

jbriggs444, vanhees71 and topsquark
I'll have a go at this.
The time between two events measured in a frame where the events happen at the same position is called the proper time between the events.
The coordinate time is measured in a frame where the two events have a change in position. The coordinate time is usually related to the proper time, because the proper time is invariant in all inertial frames.

The proper time between two events for a non-inertial observer is generally less than it is for an inertial observer (ie the twin effect where the two events are departure and reunion).

Please correct me if I'm wrong.

DAH said:
The time between two events measured in a frame where the events happen at the same position is called the proper time between the events.
Not quite. Proper time is measured along a worldline, and is independent of any choice of frame. But if you choose a frame properly, you can have coordinate time match proper time along your chosen worldline. (Note that if your chosen worldline is non-inertial, the frame you choose will have to be a non-inertial frame.)

DAH said:
The coordinate time is measured in a frame where the two events have a change in position.
No. Coordinate time is just a coordinate. There is no requirement that the two events must be at different spatial positions for their coordinate times to be defined.

DAH said:
The proper time between two events for a non-inertial observer is generally less than it is for an inertial observer (ie the twin effect where the two events are departure and reunion).
Assuming both observers' worldlines pass through the same events, the inertial observer's proper time will always be the longest in flat spacetime (i.e., in special relativity).

DAH, vanhees71, PeroK and 1 other person
PeterDonis said:
Not quite. Proper time is measured along a worldline, and is independent of any choice of frame. But if you choose a frame properly, you can have coordinate time match proper time along your chosen worldline. (Note that if your chosen worldline is non-inertial, the frame you choose will have to be a non-inertial frame.)No. Coordinate time is just a coordinate. There is no requirement that the two events must be at different spatial positions for their coordinate times to be defined.Assuming both observers' worldlines pass through the same events, the inertial observer's proper time will always be the longest in flat spacetime (i.e., in special relativity).
Thanks for the corrections.
Yes I should have added in worldline for clarity.
TBH here in the UK I don't think we use the term coordinate time, Its like you said, just the time coordinate which could be the proper time depending on your worldline.
I'm starting a module this year on SR and GR so you might see more of me in here as the weeks go by.

Dale
I always like to maintain a distinction between proper time and coordinate time even when using a coordinate system that matches some clock's proper time. The distinction is that proper time is defined only on the clock's worldline and it is invariant. The coordinate time is defined everywhere and it is not invariant. Basically, they are functions with different domains, even though in the intersection of their domains they do match.

PeterDonis, PeroK, DAH and 1 other person
DAH said:
TBH here in the UK I don't think we use the term coordinate time
We do. Or at least I was taught the term twenty years ago.

DAH
Ibix said:
We do. Or at least I was taught the term twenty years ago.
I think there's a few minor differences with some terms and notation but I'm pretty sure the actual physics will be equally good.

Thank you everyone.

## 1. What is a non-inertial frame?

A non-inertial frame is a reference frame that is accelerating or rotating. In this type of frame, objects appear to experience a force even if no external force is acting on them. This is due to the fact that the frame itself is accelerating or rotating, causing the laws of motion to appear different.

## 2. How does time behave in a non-inertial frame?

In a non-inertial frame, time can appear to be distorted or non-uniform. This is because the acceleration or rotation of the frame can affect the rate at which time flows. This phenomenon is known as time dilation and is described by Einstein's theory of relativity.

## 3. What is the connection between gravity and non-inertial frames?

Gravity is a force that can cause objects to accelerate, and therefore can create a non-inertial frame. In fact, the concept of a non-inertial frame was first introduced by Sir Isaac Newton to explain the effects of gravity on objects in motion.

## 4. How does the concept of time in non-inertial frames relate to the theory of general relativity?

The theory of general relativity, developed by Albert Einstein, explains gravity as the curvature of spacetime. In this theory, the presence of mass or energy can cause spacetime to curve, which in turn affects the flow of time in a non-inertial frame. This is known as gravitational time dilation.

## 5. Can the effects of non-inertial frames on time be observed in everyday life?

Yes, the effects of non-inertial frames on time can be observed in everyday life. For example, GPS satellites must take into account the time dilation caused by their high speeds and the Earth's gravitational pull in order to accurately calculate and transmit location data. This is just one example of how the concept of time in non-inertial frames has practical applications in our modern world.

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