We often say that "time seems to be going very fast" or "it feels like the time just sped up". When you think about it, do these phrases make any sense? Isn't our definition of speed "how far some distance passes in a certain amount of time"? Doesn't that make it incorrect to say "the speed of time", because time does not travel any distance? Or does it?
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Isn't our definition of speed "how far some distance passes in a certain amount of time"? Doesn't that make it incorrect to say "the speed of time", because time does not travel any distance?
For example, time is often defined operationally in high precision clocks with reference to the rate at which radioactive elements decay or change electromagnetic states without moving any measurable distance.
Distance per time is the definition of speed which is related to time (velocity is speed associated with a particular direction of travel), but is not by any means the only way or most fundamental way to define time.
In general relativity, time is a dimension similar in many respects to length. It is a form of distance rather than something defined by distance.
When you think about it, do these phrases make any sense? Or does it?
The phrase refers to the psychological perception of time, rather than to the physical reality. Sometimes time perceptually seems to go faster or slower in particular circumstances, even when the rate at which it passes doesn't actually change.
There are circumstances when time for one observer can slow down as a physical matter relative to the way that it is experienced by other observers as a result of special relativity. Thus, an observer who spends a long time on a round trip at close to the speed of light will have experienced less physical time elapse than one who stayed at the point of origin/destination the whole time. So, the concept of time flowing at different rates for different people is not a nonsensical one. But, that is not what the expressions you reference are referring to (at least historically).
Velocity (speed) is distance travelled divided by the duration of time spent travelling. V = d/t
What distance does time traverse that can be measured in other terms than duration itself?
If you use t for d, you get V = t/t.
This means that the speed of time is 1 second per second.
Is this a helpful value?
Or are you talking about perceived duration versus measured duration?
If, on the other hand, you're using time metaphorically, then there are other possibilities,
Time Is Money metaphor theme.
It may be apocryphal, but there's an Einstein quote,
A minute with your hand on a hot stove feels like an hour. An hour with a beautiful woman feels like a minute. That’s relativity.
Such expressions as 'time flies' are purely subjective and are metaphors. I am neither qualified nor keen to stray into the realm of Physics but at least I can say that time is a state, without facing ostracization from too many of the places I frequent.
Within the state of time and within the state of space, there is movement; some fast, some slow. And all the movements are relative, one to another.
My lifetime has been about twenty five thousand times as long as the fleeting moments of a mayfly's existence, and the tortoise in next door's garden could well outlive me.
But the expressions you use above, subjective as they are, deserve to be stated and should not be suppressed for merely scientific reasons.
The speed of time can be defined as the ratio between the amount of time that has passed when measured objectively, and the amount of time that seems to have passed when assessed subjectively. If three hours has passed but it seems like an instant, we say that the time went very quickly.
So it does make sense: it's a measure of how much our perceptions differ from reality.
Whether there's any scientific basis to it, of course, I can't tell. If there is then it's in the realm of psychology rather than physics.
It absolutely makes sense, even if in a metaphorical way.
The most concrete meaning of "speed" is, as you say, distance per time. For other kinds of change per unit time, we usually say "rate" instead, but "speed" is easily understood. You can find examples of people referring to things like "the speed of temperature change" on the web, and, even if you think "speed" is not the best term, it makes sense as a term for the change in something per quantity of time.
But then you can take another metaphorical step. In a literal sense, if time is just a single thing, then change in time per change in time will always be 1. But, if there can somehow be two kinds of time, then the rate of one in terms of the other makes perfect sense. There can be "time as perceived" as compared to "time as measured", which comes up in the examples you gave. There can also be "time measured by one observer" compared to "time measured by another observer with a different velocity", which can literally be different, although time is not generally referred to has having a speed in that context.
Btw, in case anyone's interested in the real speed of time, then time moves at the speed of light: that is to say, we are all travelling from the past to the future at light speed. This is why travelling at physical great speed makes time slow down: we are already at the max speed for the universe, so if we want to travel in space in addition to that, we need to use some of our "time speed" to travel in that extra dimension.
Out of context it is difficult to say if "the speed of time" makes any sense. In your other examples "time seems..." and "it feels like..." it is clear that they express the idea that there is a subjective perception of time that differs from its (assumed) absolute nature.
My favourite expression in this category: time flies when you're having fun!
I thought that one hour had passed, but looking at the clock it was two hours. Real time went twice as fast as I thought. The speed of time was two real hours per imagined hour.
The time itself to do something could be seen as an absolute. If it takes less time to do the same thing one day, that absolute time feels like it passed more rapidly; if it takes longer, that time seems like it was longer - which indeed it was.
It's only a metaphor, referencing our feeling of a period of time passing. 'Seems like months since I last saw you'. 'I'll only be a minute'. There are plenty of phrases we use referring to the passing of time that don't hold water. But we keep on using them, as they convey our thoughts quite well !
If you're sitting in a train, and you watch the trees and houses pass you, would you say they're whizzing by quickly? I would, even if I'm aware that it's the train that is moving. (Not that there's much a difference in classical kinetics.)
While it's commonly believed that it's our minds that move through time - although this isn't exactly a rigorous statement - the idioms you mention are consistent with other ways to describe similar phenomena that are just artifacts of our perception rather than "real" according to some model of physics. (Compare and contrast: how humans don't have a sense for anything we call "wet.")
Heres a thought experiment: 1) We divide a homogenous piece of radium into two equal masses.
2) We verify that their radiation is the same: They release the same amount of radiation per second.
3) We now spin one of the pieces EXTREMELY FAST ! (To a "relativistic" speed!)
If we now measure and compare the radiation from the spinning piece of radium with its stationary twin we will find that the spinning piece is releasing less radiation per second ...
And the physical explanation for this is that the speed of time in the spinning radium is slower that in its stationary twin.
ONE SECOND PASSED INSIDE THE STATIONARY TWIN IS OF SHORTER DURATION THAN A SECOND IN ITS SPINNING TWIN!
Isn't our definition of speed "how far some distance passes in a certain amount of time"?
Ah! There's your gaffe. A good dictionary will list multiple meanings of the word speed – some as nouns, and some as verbs. Some of these definitions will be technical:
PhysicsDistance traveled divided by the time of travel.
while others will be more figurative or metaphorical:
speed (v.) To pass quickly : The days sped by. The months have sped along.
As for the phrase you ask about – the speed of time – I wonder: What do you mean by "incorrect"? What might be incorrect on a science test could well be grammatically correct in an essay, and could even be profoundly correct in a poem. In the English language, the issue is not whether or not time travels, but how much words bend.
Our fleeting lives accelerate
In ways that are sublime
So thus the elders ruminate
And scrawl a simple rhyme
The physicists pontificate
A constant paradigm
While linguists pause to estimate
The velocity of time
If this wasn't the answer you wanted, then (as your first commenter said) you've probably asked on the wrong Exchange.