The pickle in the ice cream for 19th century physicists was light. Or, to use my other unnecessary metaphor from The Genesis of Supersymmetry, Electromagnetics was the evil Serpent in the mechanical Garden of Eden; Newton and Maxwell did not agree, light did not act as matter should act. Einstein reconciled them by showing that space and time are relative, and that the speed of light and the laws of physics are the same for everyone, regardless of their motion, like George FitzGerald and Hendrik Lorentz said - because energy and mass are "converted" into each other, which keeps the speed of light constant.
Instead of Einstein or Lorentz, "spacetime" was instead the brainchild of mathematician Hermann Minkowski, who riffed on fellow math guru Henri Poincaré's mathematical wizardry using a Lorentz transformation in terms of Euclidean rotation. Minkowski said since the space is then a pseudo-Euclidean space, that rotation is a representation of a hyperbolic rotation - and then he restated Maxwell equations in four dimensions, which was pretty cool.
But in 1908 he really got going and did his magic by reformulating Einstein's new special relativity in four dimensions, declaring, "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." (1)
Time dilation experiment finds that she will be 90 billionths of a second older over a 79-year lifetime. Link: Daily Mail, because who else would have this picture? Credit: Corbis
Since then, there has been a lot of explaining but time and spacetime remains so baffling that it can be argued that time is not mathematically represented by an imaginary coordinate at all and is not even a fourth dimension of space but that space and time are two separate entities. (2)
When it comes to talking about time as a physical dimension, like with clocks, things are no easier. How old are you? Well, it depends on which epoch you were born in. The spin of the Earth has slowed down through the millennia and therefore so has time.(3) The same 19th century that gave us those breakthroughs in electromagnetics and relativity gave us a relative definition of time that we now know just doesn't cut it: we think in terms of 60 seconds in a minute and 60 minutes in an hour and 24 hours in a day without considering that the day itself has changed.
Earth is turning on its axis 2/1000ths of a second slower than it did in the 19th century. Why does it matter? Well, it doesn't if you are going to a party but, as Ivan Amato, a freelance science writer who runs DC Science Cafe, put it in the Washington Post, modern GPS systems and those speed-of-light radio signals travel about one foot every every nanosecond - so if their synchronization is off by 2/1000ths of a second, your GPS couldn't tell you if you were in Boston or Chicago.
Now imagine heading to Mars. Microsecond changes in the speed of Earth’s rotation could have thrown the navigational precision of the Curiosity Rover off by a mile or more, which might have meant disaster. The pull of the Moon is the biggest reason our day is a different length than the days Einstein had but that's not all, natural disasters matter too. A team at JPL computed that the Sumatran earthquake of Dec. 26, 2004 moved enough of the Earth’s mass to change the length of an Earth day by almost 7 microseconds.
No wonder Harold Camping was so spectacularly wrong when he tried to calculate the day the world would end - the world had completely different day lengths than when the Bible was written. Amato also interviewed Dr. Thomas O’Brian, chief of the Time and Frequency Division of the National Institute of Standards and Technology (NIST) and Ph.D. in experimental atomic physics from the University of Wisconsin at Madison, who noted that 200 million years in the future we'll have 25-hour days and 335-day years - time will have slowed down a lot compared to dinosaurs, who had 385 days that were 23 hours long in each year. A 40-year-old man today will only be 36 then!
It's not the coolest thing about time and relativity, of course - 200 million years is not a practical concept - the fact that scientists can measure how taller people are aging different than short people is.
(1) That really didn't happen. Einstein dismissed the "superfluous learnedness" of Minkowski (you know, like when you read arXiv papers from someone claiming they have mathematically created time travel) but he did take advantage of the tensor mathematics that made spacetime possible when he began deriving his theory of general relativity.
(2) Amrit Sorli and Davide Fiscaletti. “Special theory of relativity in a three-dimensional Euclidean space.” Physics Essays: March 2012, Vol. 25, No. 1, pp. 141-143. DOI: 10.4006/0836-1398-25.1.141
(3) It even befuddled the Nobel committee. When nominations for the 1912 Nobel prize rolled around, the debate was whether or not to award a prize for the Theory of Relativity jointly to Lorentz and Einstein or just Einstein, because Einstein put it together but Lorentz had done the early math.
They settled the problem by giving the award to Nils Gustaf Dalen for the "invention of automatic regulators for lighting coastal beacons and light buoys during darkness or other periods of reduced visibility", which shows you that the Nobel Physics committee was as clueless in 1912 as the Peace committee is...well, every year.
The special and general theories of relativity never got a Nobel prize for anyone.
Gravity Probe B - Stanford University
Physicists continue work to abolish time as fourth dimension of space Lisa Zyga, PhysOrg
"Spacetime has No Time Dimension" -- New Theory Claims that Time is Not the 4th Dimension - The Daily Galaxy