The Dow Jones Industrials (^DJI 0.76%) managed to bounce back a little on Friday, rising almost 45 points, and closing out what was shaping up to be a bad week on a somewhat positive note. Yet, as the small-cap sector of the stock market has underperformed the Dow substantially, some investors wonder whether the Dow is truly representative of the broader stock market. Indeed, even though the Dow purports to be an industrial index, it lacks even one representative from the iconic American industrial sector: automotive manufacturing. Eventually, the Dow Jones Industrials will need to admit an automaker to its ranks, and when it does, which will it be: General Motors (GM 0.92%), Ford (F 1.23%), or Tesla Motors (TSLA -2.07%)?

Source: General Motors.

From an historical perspective, General Motors is the stock with the best claim to be in the Dow. After all, General Motors was a Dow component between 1925 and 2009, until the iconic automaker declared bankruptcy in the aftermath of the financial crisis. With a market capitalization of just $54 billion, General Motors would be among the smallest companies if it were admitted to the Dow Jones Industrials, and its share price of $34 would make it a relatively lightweight component in calculating the Dow's value.

Now that the government has divested its stake in General Motors, the managers who choose the Dow's components might be more willing at least to consider the move; but with a huge amount of potential liability hanging over the company as a result of recent recall scandals, General Motors shouldn't expect any sort of invitation until it can resolve troubling allegations about its behavior concerning alleged defects in its automobiles.

Source: Ford.

Ford also has plenty of arguments for deserving to be in the Dow. It's the only major automaker not to have taken a bailout during the financial crisis, and it avoided bankruptcy, as well. Ford's market capitalization is the largest of any automaker in the U.S. and, along with General Motors, Ford has a healthy dividend yield of more than 3%. Fundamentally, Ford's business continues to fire on all cylinders, with a host of new product rollouts coming in the next year that could carry the automaker into its next generation of success. Yet, from a feasibility standpoint, Ford's biggest obstacle in joining the Dow Jones Industrials is its share price, which at less than $16 per share, would give it almost no influence in the Dow at all.

Source: Tesla Motors.

Finally, Tesla Motors might not be in a position to join the Dow anytime soon, as it hasn't even made it into the less-exclusive ranks of the S&P 500. Given enough time, though, Tesla could well beat out its automaker peers to join the Dow, especially if its expansion plans go beyond the automotive industry to include broader applications for battery technology. Tesla's planned gigafactory could help reduce battery-cell costs, not only for use in Tesla Motors' vehicles, but also for storage of electricity collected from solar panels and other as-yet unknown uses. As a key input to electric-vehicle production, though, boosting battery production could enable Tesla to ramp-up overall vehicle throughput, putting it on a faster course to match Ford and General Motors, and potentially to join the Dow Jones Industrials in the long run.

The Dow Jones Industrials have gone without a car company for five years, and they can afford to go without one for a little longer. But in the end, it's likely that Ford, General Motors, or Tesla Motors will eventually get a Dow invitation, if only because an American stock market gauge without an American institution in the auto industry just doesn't make sense.