@ Everybody but mostly Snake:
Some details on the Chancellorsville Campaign and the Wilderness Campaign are in order. Courtesy of
Chancellorsville, Stephen Sears, and
The Battle of The Wilderness, May 5-6, Gordon C. Rhea.
Hooker certainly "stole a march" on Lee, if by "stole a march" you mean moved troops from point A to point B without Lee being aware of it. That much is certainly true. The details however are important. In this case, point A is the Union Army's winter quarters to the northeast of Lee. And point B is the area from Ely's Ford to Chancellorsville proper, northwest and west of Lee. Which one depends on how you define Lee being aware of it.
Its value for the current discussion is minimal. Using both straight-line distance and marching distance, Lee's army is closer to Richmond than both point A and point B. Furthermore, he controls the RF&P railroad thus increasing this advantage.
(I would love to talk about the 10 or so different lucky breaks that Lee got during the Chancellorsville Campaign - by all rights Hooker should have won, and it was *minimally* his fault, but they are not relevant to this discussion).
With Grant the situation is even more interesting. His winter camps lay between the Rappahannock and Rapidan Rivers, while Lee's lay significantly far to the west. In terms of distance to Richmond, Lee's distance was slightly closer. To start.
It is true that Grant never planned to battle in the Wilderness. However, his original plan was *NOT* to move south of it. He planned to turn west and go up directly against Lee. He hoped, ideally, that Lee would take up a defensive position at or near Mine Run, as he had done late in 1863. Its north flank would be against the Rapidan River, but its southern flank would be more open. Grant then planned to turn said flank utilizing his 2:1 numerical advantage. Tactically this would place Grant minimally closer to Richmond than Lee. Strategically once battle had joined the distances would be equal.
Furthermore Grant calculated how quickly Lee would move from his Winter Camps to the Mine Run line, based on the aforesaid 1863 campaign. The AoTP and the IXth corps were moving behind schedule, but Grant had never intended to beat Lee to the Mine Run line. The slowness of the Union army became important only when Lee attacked Grant. Also it should be pointed out that the IXth corps did not get across the Rapidan River until May 6th, halfway through the battle, and elements of the VIth corps also arrived late on the 5th. So *no*, in fact Grant did not have his entire army across the river before Lee had already started the battle, let alone been prepared to.
The important points of all the above: with the Wilderness Campaign, for a short time on May 4th Grant was closer to Richmond than Lee. But Lee was entirely aware of this fact and moving to attack Grant at that moment, while Grant was never intending to move directy on Richmond in the first place.
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Now let's compare to the hypothetical.
Assumption 1: Burnside intends to move to Richmond, with the intent of capturing it, as quickly and directly as reasonably possible. "Lee's Army" is not the primary objective in late 1862. Knowing Burnside, "quick" means anything but, but "directly" means a straight line if at all possible, and Burnside certainly can aim for that.
Assumption 2: The pontoon bridges arrive on time. This is the hypothetical assumption.
Marye's heights and surrounding territory is an obvious position for anybody, so we can assume that Burnside will assemble his army there. Considering that iOTL, Franklin's Grand Division crossed in a single day with no problems, unopposed, we can assume that with twice (or more) the # of pontoons, it will take Burnside no more than 2 days to cross and assemble his full army on the southern bank of the Rappahannock River. This assumes that they are unopposed, but as others have mentioned there was only 1 Confederate regiment in the area, acting only as survelliance, so yes, I argue this qualifies.
Another point of clarification: By crossing at Fredericksburg, the word "Rapidan" should *NOT* enter this discussion. The Rapidan is a tributary of the Rappahannock, that enters it above Fredericksburg, so if Burnside crosses at Fredericksburg he crosses both rivers at once, they being the same river.
Now let's talk about geography. From Fredericksburg, the direct route to Richmond is pretty much due South. This route can be done, using any of the following: Telegraph Road, the RF&P Railroad, Bowling Green Road. All of these will at least get you to the North Anna River, and all of these go directly south.
Also from Fredericksburg, the route to get into the Wilderness is due West. Using the Orange Turnpike and/or Orange Plank Road. To enter the Wilderness, starting from Fredericksburg, requires at least a 5 mile march.
(Now for some geometry

)
We can call this 5-mile line segment W-F. Burnside does not intend to do this, but instead march to Richmond, the longer segment F-R. The angle W-F-R is a right angle. Therefore, the instant Burnside leaves Fredericksburg en route to Richmond, his distance to the Wilderness will increase. And it will continue increasing until he reaches Richmond or is otherwise stopped.
In particular Burnside's distance to the Wilderness will never be equal to 0. This includes the case where Burnside does not move at all beyond Fredericksburg, because in this base case he is still 5 miles to the east of it.
Therefore, Burnside will at no point in this campaign ever be in the Wilderness. Q. E. D.
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Now let's throw Lee into the picture. According to various other posters, and I believe them for the moment, at the instant the pontoon bridges are set up in this hypothetical the nearest large Confederate force is Longstreet's corps, which they say is a 4 day march to the west. A lot of people have been saying that Burnside will cross his entire army over the river in 1 day. I say that pontoon crossings are a tricky affair, given also supply trains, so I give him 2 days.
In the best case for Lee he knows the instant that Burnside starts crossing the river and immediately moves his men to intercept him. Neither of these are necessarily true statements, but we'll assume them for now.
Therefore, by the time Burnside has his entire army across the river, Longstreet's corps, half of Lee's army, is still 2 days march to the west of Fredericksburg.
Now from Burnside's perspective, the whole point of this campaign was to get to Richmond before Lee can intercept him. The move to Fredericksburg in the first place was supposed to be unexpected and gain him a temporary time advantage, which it succeeded in. However slowly Burnside will move, I argue that he *WILL*, in fact, move towards Richmond. Sitting in Fredericksburg accomplishes nothing, and is contrary to his strict campaign plan, as approved by Lincoln (apparently an important point to Burnside).
A fair-to-poor day's march for Union troops is ~10 miles. Less than that is definitely poor, or at least leisurely. Burnside may not be able to organize the march well (maybe), but he certainly is not *trying* to be leisurely. Again, that would go against the whole goal of getting to Richmond first. So I assume he can manage 10 miles a day.
The marching distance from Fredericksburg to the North Anna River is ~25 miles. So in 2 days Burnside will definitely be almost there. 2.5 days and he will be there.
Now let's make another geometry figure. The triangle LFN. L-F-N is a right angle, again centered at Fredericksburg. N is the crossing sites of the North Anna River due south of F. L is Longstreet's position at the specific time that it would take him 2 days to traverse the line L-F.
It will, we have seen, take Burnside 2.5 days to traverse the line F-N. During this time he is travelling in a straight line.
Longstreet would like to intercept Burnside. The distance he has to cover to the east will take a minimum of 2 days, by which time Burnside has almost reached N. Longstreet could, at this point, reach F, but that serves little purpose. (Actually, cutting Burnsides supply lines at this point....an interesting idea. One that should be discussed. But not the topic at hand).
If Longstreet adds on any southward distance he increases the time it takes him to make the march. Because this is the hypotenuse it does not add on time to the same order of magnitude. Still, to travel the line L-N should take absolutely no less than 2.5 days. Probably more, but we can go with 2.5.
Note that that is the same amount of time it takes Burnside to get to N. And the route Longstreet would take would involve the Virginia Central railroad, thus placing him on the south bank of N.
In other words, assuming that Lee's men (Longstreet's) march as fast as can reasonably be expected, and Burnside's march as slowly as can be reasonbly expected, Lee will still not catch up to Burnside until the North Anna River.
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Of course, once the North Anna is reached, Lee has every possibility of beating Burnside in detail. If nothing else, the OTL inverted V should do quite nicely. Certainly should buy another week or so to bring up Jackson's corps.
On the other hand, Burnside will start with a 4:1 superiority. And Lee will not have any pre-made entrenchments. And the inverted-V is not an obvious position. iOTL it took Lee 2+ days to determine it and put his army into said position. So perhaps Burnside would even win such a fight. Or perhaps not, he is Burnside after all, and Franklin sucks as a wing commander. But it is certainly a discussable topic.
However, the bottom line is, there will be NO fighting BEFORE the North Anna River, and words such as "Rapidan" and "Wilderness" will have no role to play in the campaign.