Kinetic vs Gravitational Time Dilation: Perceived Speed

[Involute]

TL;DR Summary

It seems to me that an observer perceives relativistic speed of an object differently under each of these forms of time dilation. This is probably due to my misunderstanding. Hoping someone can straighten me out.

As an object approaches a black hole’s event horizon, it experiences increasing gravitational time dilation, causing it to appear to an outside observer to slow down, until, at the event horizon, it appears to stop. An object traveling in space that increases its velocity from one non-relativistic velocity to another will be observed to accelerate. If it were able to increase its velocity to, say, 99% the speed of light, wouldn’t kinetic time dilation cause any further increase in velocity appear as a deceleration? If so, at what velocity does an increase in velocity change from producing an observed increase in velocity to producing an observed decrease in velocity?

 

[PeroK]

An object traveling in space that increases its velocity from one non-relativistic velocity to another will be observed to accelerate. If it were able to increase its velocity to, say, 99% the speed of light, wouldn’t kinetic time dilation cause any further increase in velocity appear as a deceleration? If so, at what velocity does an increase in velocity change from producing an observed increase in velocity to producing an observed decrease in velocity?

There is no point at which acceleration of an object in motion is observed as deceleration. I'm not entirely sure why you would think that it might. To be specific:

For one-dimensional motion, where an object has a proper acceleration of $a$ (i.e. acceleration as measured in an inertial reference frame where the object is instantaneously at rest), the acceleration as measured in a frame where the object has speed $v$ is: $$a' = \frac 1 {\gamma^3} a$$ where $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$.

 

[Dale]

In addition to @PeroK ‘s comment above, you should distinguish between coordinate acceleration and proper acceleration.

Coordinate acceleration is the second derivative of the spatial coordinates with respect to the time coordinate along some worldline. Obviously, this requires the specification of a coordinate system, which you didn’t do.

Proper acceleration is the acceleration measured by an accelerometer on some worldline. This is coordinate independent so all observers agree on it at all points, including near the event horizon.

 

[PeterDonis]

It seems to me that an observer perceives relativistic speed of an object differently under each of these forms of time dilation.

That's because these different forms of time dilation occur in different spacetime geometries. The spacetime geometry makes a big difference.

 

[pervect]

Summary:: It seems to me that an observer perceives relativistic speed of an object differently under each of these forms of time dilation. This is probably due to my misunderstanding. Hoping someone can straighten me out.

The following may or may not be helpful to you in figuring some things out.

From Baez's "The Meaning of Einstein's Equations", https://arxiv.org/abs/gr-qc/0103044. (See the pdf for the full text version, this links to the abstract).

Before stating Einstein’s equation, we need a little preparation. We assume the
reader is somewhat familiar with special relativity — otherwise general relativity
will be too hard. But there are some big differences between special and general
relativity, which can cause immense confusion if neglected.
In special relativity, we cannot talk about absolute velocities, but only rel-
ative velocities. For example, we cannot sensibly ask if a particle is at rest,
only whether it is at rest relative to another. The reason is that in this theory,
velocities are described as vectors in 4-dimensional spacetime. Switching to a
different inertial coordinate system can change which way these vectors point
relative to our coordinate axes, but not whether two of them point the same
way.
In general relativity, we cannot even talk about relative velocities, except for
two particles at the same point of spacetime — that is, at the same place at the
same instant. The reason is that in general relativity, we take very seriously the
notion that a vector is a little arrow sitting at a particular point in spacetime.
To compare vectors at different points of spacetime, we must carry one over to
the other. The process of carrying a vector along a path without turning or
stretching it is called ‘parallel transport’. When spacetime is curved, the result
of parallel transport from one point to another depends on the path taken! In
fact, this is the very definition of what it means for spacetime to be curved.
Thus it is ambiguous to ask whether two particles have the same velocity vector
unless they are at the same point of spacetime.

So, you are asking about relative velocities, which are well defined in special relativity, but not so well defined in general relativity. To try to be clear, relative velocities are well defined between two objects at the same location, but the relative velocity of a distant object is ambiguous. So one can always talk about the velocity of an object relative to that of a static observer, as a static observer is a family of objects that are defined at every possible location and time.

The path forwards, though, is a long one. But the starting point is to understand SR before GR, and to understand how SR is a 4 dimensional theory. Understanding space-time diagrams and appreciating that they are useful and necessary and should be learned - that they are not not just makework - is perhaps an equivalent alternative.

Then one can come to a general understanding that in GR, one might draw a 2 dimensional space-time diagram (with one spatial dimension and one time dimension) on a curved surface, such as a sphere or a football, rather than on a flat sheet of paper.

And one can appreciate then that the relative velocity of two objects is an "angle" on the space-time diagram. A bit of understanding spherical trignometry can be helpful, too, in getting some crude ideas of what happens with curved spatial geometires, then applying this understanding of curved spatial geometries to curved space-time geometeries.

 

[LBoy]

1. An object traveling in space that increases its velocity from one non-relativistic velocity to another will be observed to accelerate. If it were able to increase its velocity to, say, 99% the speed of light, wouldn’t kinetic time dilation cause any further increase in velocity appear as a deceleration?

2. If so, at what velocity does an increase in velocity change from producing an observed increase in velocity to producing an observed decrease in velocity?

1.

No. The constant acceleration observed in our inertial system causes a constant increase in velocity. According to STW - in this case time dilation is increasing. This situation can continue indefinitely - acceleration will cause an increase in velocity and increasing time dilation, although from our point of view the increase in velocity will be slower and slower.

There will be no deceleration (which requires a change in the direction of the force) - there will only be a decrease in the speed of velocity increase.

Example of decreasing speed of increasing velocity:

You are driving a car and in 1 minute your velocity increases 0 from 0 to 100 kilometers per hour.
During the next minute your speed increases from 100 kilometers per hour to 150 kilometers per hour.

During the first minute, your speed has increased by 100 km/h. During the second minute, your speed has increased by 150 - 100 km/h i.e., by 50 kilometers per hour. That is, the speed of increase in velocity has decreased - but your velocity continues to increase.

2.

At none. Increasing velocity will produce decreasing speed of increase but not decrease of velocity itself. Velocity will increase, however the velocity of light will not be achieved. And there will be no deccelaration at any point.